Grade 10 / General information on functions 90 exercises (100% corrected)

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ChingQuizz : 7 exercises available for Quizz assessment : 1. Introduction to functions E. 8070 A company manufactures pulleys used in the automotive industry. We assume that all production is sold. The company can manufacture between 0 and 3 600 pulleys per week. Let x be the number of thousands of pulleys man-ufactured and sold in a week. ( x therefore varies between 0 ; 3 ; 6 ) . The weekly profit is denoted by B ( x ) and is expressed in thousands of euros. Below, we have represented the function B on a coordinate plane. Each result will be given to the nearest hundred pulleys or to the nearest hundred euros, as appropriate. The lines useful for understanding the reasoning will be left on the graph, and a written answer on the exam paper will be expected for each question asked. 1 What is the profit made by the company when it pro-duces and sells 2 400 pulleys per week? 2 How many pulleys must the company produce and sell in order to make a profit of 13 000 euros? E. 1348 The graph below represents the value of the CAC 40 (stock market indicator on forty compa-nies on the Paris stock exchange) 1 We’re interested in the day of July 2, what was the value of the CAC 40: a à 0h? b à 6h? c à midi? d à 18h? 2 On the chart, at what point did the CAC 40 have : a une value of 4200? b a value of 4300? 3 Choose from the following two sentences the correct one : a ˇ This graph gives the date as a function of the value of the CAC 40 ı b ˇ This chart shows the value of the CAC 40 as a func-tion of date ı https://chingmath.fr chapExoCorrec/8070 sacados/8070 x00,20,40,60,811,21,41,61,822,22,42,62,833,23,43,6y-1123456789101112131415 chapExoCorrec/1348 sacados/1348 1/072/073/074/075/076/074100420043004400
E.7049 A patient is injected with a drug and the concentration, in grams in liters, of this drug in the blood is measured regularly for 15 hours. The curve shown below is obtained : With the precision permitted by the graph, indicate : the concentration at the initial instant ; the time interval during which the concentration is greater than or equal to 0.4 gram per liter. The necessary construction lines will appear on the graph. E. 1764 In the reference frame shown below, consider the representative curve C of the function f : 1 Place the point A ( − 1.5 ; 2.5) . 2 Consider the following points in the plane : B ( − 2 ; 3) ; C (2.5 ; 1) ; D (0.5 ; − 1) ; E (0.25 ; 0.5) a Place these points on the marker. b Which of these points definitely belong to the curve C . 3 Place the single point F belonging to the curve C having − 1 as its abscissa. Give its coordinates. 4 How many points on the curve C have the ordinate value 1? Specify the coordinates of these points. E.4648 Consider a glass with a rectangular parallelepiped base, the top of which is the base of a square pyramid. Note h the height of the liquid contained in the glass : Which of the three curves below represents the volume of liq-uid V as a function of height h ? 2. Introduction to functions: table of values E.1763 Onagre is a mobile phone operator offering the following subscriptions : Subscription A: subscription fee of ¿ 19, then ¿ 0.30 per minute of communication ; Subscription B : subscription fee of ¿ 29, then ¿ 0.20 per minute of communication. Complete the following table : Duration (in minutes) 30 45 60 90 Plan A (in euros) Plan B (in euros) https://chingmath.fr chapExoCorrec/7049 sacados/7049 Temps (en heure)012345678910111213141516Concentration (g/‘)0,20,40,60,811,21,41,61,822,2 chapExoCorrec/1764 sacados/1764 x-4-3-2-101234y-2-1123C chapExoCorrec/4648 sacados/4648 hh hVCfa hVCgb hVChc chapExoCorrec/1763 sacados/1763
(excerpt from the Guadeloupe patent, June 2006.) E.8020 A company wants to install a garden on either side of the entrance path to its warehouse. The garden is shown in dot-ted lines in the diagram op-posite. The diagram opposite shows the dimensions of the entrance to the ware-house. The quadrilateral ABCD is a rectangle. 1 Taking 30 m as the width of the warehouse entrance at road level ( PQ =30 m ) . a Determine the area of the access road from the road to the warehouse entrance. b Determine the total area of the garden. 2 Complete the table of values below : PQ (in m ) 10 20 30 35 Area of the garden (in m 2 ) 3 How can we justify that it is possible to choose the size of the driveway entrance (the distance PQ ) so that the total area of the garden is 600 m 2 . Reminder : 3. Introduction to functions: algebraic expressions E.8021 Here’s a script entered by Al-ice into an algorithmic program : 1 Alice has chosen 3 as her number, calculate the value of ˇ Result 1 ı. 2 Generalization: a Calling x the number chosen in the algorithm, give a literal expression translating the result corresponding to Alice’s algorithm. b Find the number or numbers chosen by Alice that cor-respond to the resulttat displayed below : E. 11373 Here are two calculation pro-grams : Program A Choose a starting num-ber Subtract 1 from the chosen number Calculate the square of the difference obtained Add twice the starting number to the result Write down the result obtained Program B Choose a starting num-ber Calculate the square of the chosen number Add 1 to the result Write the result ob-tained 1 With program A and choosing 3 as the input number, what is the value of the output? 2 With program B and choosing 3 as the input number, what is the output value? https://chingmath.fr chapExoCorrec/8020 sacados/8020 ABCDMNPQ8m40m30m largeur(‘Longueur(LRectangleALבpetitebase(bgrandeBase(Bhauteur(hTrapèzeABb×h2côté(cCarrédAc2ABCTrianglerectangleAAB×AC2base(bhauteur(hTriangleAb×h2rayon(rDiamètre(DCercleP2×ı×rAı×r2 chapExoCorrec/8021 sacados/8021 quandest cliquédemanderChoisissez un nombreet attendremettreNombreàréponsemettreRésultat 1à2*Nombre+3mettreRésultat 1àRésultat 1*Résultat 1direregroupele résultat 1 estRésultat 1 le résultat 1 est9 chapExoCorrec/11373 sacados/11373
E.11374 Here are two calculation programs : Program A Choose a starting num-ber Subtract 1 from the chosen number Calculate the square of the difference obtained Add twice the starting number to the result Write down the result obtained Program B Choose a starting num-ber Calculate the square of the chosen number Add 1 to the result Write the result ob-tained 1 Show that when the starting number is 3 , the result ob-tained with the program is A . 2 When the number of starts is 3 , what result is obtained with the program B E. 11375 Consider two real numbers x and y linked by the relation: y =2 × x − 5 We then say that ˇ y is given as a function of x ı. E. 11376 The relationship below gives the value of y as a function of x : the two numbers x and y are linked by the relationship : y = x 2 − 2 x Complete the table below : x − 2 0 2 2 5 ; 4 y 4. Functional ratings E. 11382 Here are excerpts from a computer program that takes an input value and returns a return value : Let f be the function associated with this algorithm. Com-plete the dotted lines below : a f (1) = : : : b f (6) = : : : c f (7) = : : : E. 11380 Below are the first regular poly-gons : Let f be the function that associates the number of sides with the angle at the center between two consecutive vertices : Complete the dotted lines : a f (3) = 120 b f (4) = : : : c f (5) = : : : d f (6) = : : : e f (7) = : : : f f (8) = : : : E. 11381 Below is a step-by-step guide to constructing the pattern : 1 Let f be the function that associates the number of white squares in the figure with its number. Complete the dotted lines : a f (1) = : : : b f (2) = : : : c f (3) = : : : 2 We denote by g the function that associates the number of gray squares that compose it with the number of the figure. Complete the dotted lines : a g (1) = : : : b g (2) = : : : c g (3) = : : : 5. Algebraic expression: images https://chingmath.fr chapExoCorrec/11374 sacados/11374 chapExoCorrec/11375 sacados/11375 yx-201310x2×x−5 chapExoCorrec/11376 sacados/11376 chapExoCorrec/11382 sacados/11382 Valeursd’entréesValeursde sorties1346723589f chapExoCorrec/11380 sacados/11380 ¸oOTriangleéquilatéral¸oOCarré¸oOPentagone¸oOHexagone¸oOHeptagone¸oOOctogone chapExoCorrec/11381 sacados/11381 Fig.1Fig.2Fig.3Fig.4Fig.5
E.9335 Consider the following calculation program : Let f be the function that associates every number x with the output value of the calculation program when the value x is given to it. 1 Give the algebraic expression of the function f . 2 Complete the table of values below : x − 2 − 0 ; 5 1 3 ; 2 f ( x ) E.11377 Consider the following calculation program : Let f be the function that associates the input number with the output number of this algorithm. 1 Which expression defines the image of x by the function f ? a f ( x ) = x + 4 × 3 − 12 b f ( x ) = x + 4 × 3 − 12 2 Complete the table of values for the function f x − 3 − 1 2 f ( x ) 30 E. 11384 Consider the following calculation program : Let f be the function that associates the input number with the output number of this algorithm. 1 a Give the expression of the function f . b Expand and reduce the expression of the function f . 2 Complete the table of values below : x − 4 − 1 2 5 f ( x ) E.9394 For each of the functions below, de-termine the image of the number 2 : a f : x ↦−→ 10 − 2 · x 3 · x b g : x ↦−→ x 2 − 2 x + 1 E.1785 Consider the three functions f , g and h defining the image of the number x as follows : f ( x ) = 3 x − 2 ; g ( x ) = x 2 ; h ( x ) = 2 3 x − 1 Complete the following table of values : x 1.5 1 − 1 3 f ( x ) g ( x ) h ( x ) 6. Representative curve: reading images and antecedents E.273 The plane is given the reference frame shown below : https://chingmath.fr chapExoCorrec/9335 sacados/9335 quandest cliquédemanderChoisissez un nombreet attendremettreNombreàréponsemettreNombreàNombre-1mettreNombreàNombre*0,5direNombre chapExoCorrec/11377 sacados/11377 Choisir un nombreAjouter 4Multiplier par 3Soustraire 12A`cher le résultat chapExoCorrec/11384 sacados/11384 Choisir un nombreSoustraire 1Ajouter 1Multiplier les deux nombresAjouter 1 chapExoCorrec/9394 sacados/9394 chapExoCorrec/1785 sacados/1785 chapExoCorrec/273 sacados/273 x-4-3-2-101234y-2-1123ABCDEFGH
Various points have been placed in the reference frame : 1 Determine the abscissas of the following points : A ; B ; C ; D ; E Note: the answer can be written as : x A = · · · ; x B = · · · ; : : : 2 Determine the ordinates of the following points : F ; G ; H Note: the answer can be written as : y F = · · · ; y G = · · · ; : : : 3 Is it possible to determine the abscissa of the point H . E.9395 In the plane provided with the refer-ence frame below, consider the curve C representative of the function f and the two straight lines ( d ) and (Δ) . 1 Determine the coordinates of the points : A ; B ; C ; D 2 a What property characterizes the coordinates of the points on the line (Δ) ? b Complete the following sentence : All points on a vertical line have the same . . . . . . value. The straight line (Δ) has equation : x = :::::: 3 Observing the coordinates of the points on the line ( d ) , complete the following sentence : All points on a horizontal line have the same . . . . . . value. The straight line ( d ) has equation : y = :::::: E. 9337 Method : the two videos opposite illustrate the graphic reading of im-ages and antecedents. Consider a function f whose representative curve C f is given below in the reference frame below : Each answer will be justified : 1 Give the image of 4 by the function f 2 Give the antecedent of the number 1 by the function f . E.4375 Definition: In a plane equipped with a coordinate system, we call the representative curve of the function f the set of points in the plane whose coordinates are of the form ( x ; f ( x )) where the number f ( x ) is defined. In the plane with the coordinate system below, consider the curve C representing the function f . 1 Justify that the image of the number 2 , by the function f , is − 1 . Definition: Let f be a function and b a real number ( b ∈ R ) . We say that the number a is an antecedent of the number b by the function f if the image of the number a is b . That is to say: f ( a ) = b . 2 a Justify that the number − 2 ; 5 is a predecessor of the number 1 ; 5 by the function f . b Give all antecedents of the number 1 ; 5 using the func-tion f . https://chingmath.fr chapExoCorrec/9395 sacados/9395 x-4-3-2-101234y-2-1123(dCADFBCE chapExoCorrec/9337 sacados/9337 r613-1 r611-1 x-3-2-10123456y-2-112Cf chapExoCorrec/4375 sacados/4375 x-4-3-2-101234y-112Cf
E.5028 Let f be a function defined for all numbers between − 4 and 4 , whose graph is shown in the co-ordinate system below. 1 We want to determine graphically the image of the num-ber − 3 by the function f . To do this, complete the fol-lowing sentence : The line with equation . . . . . . . . . intersects the curve C f at the point with coordinates : : : ; : : : . We can deduce that the image of the number − 3 by the function f has the value . . . 2 We want to graphically determine the antecedents of the number 0 ; 5 by the function f . To do this, complete the following sentence : The line with equation . . . . . . . . . intersects the curve C f at the points with coordinates : : : ; : : : , : : : ; : : : , and : : : ; : : : . We can deduce that the antecedents of the number 0 ; 5 are: : : : : : : ; : : : : : : ; : : : : : : E.7042 A website specializes in broad- casting videos on the Internet. The site manager noticed that video loading times varied according to the number of users connected simultaneously. The aim is to estimate the loading time as a function of the number of people connected simultaneously. A function is proposed to model this situation. In the orthogonal reference frame below, we have drawn the representative curve of a function f that models the previous situation. We note x the number, expressed in thousands, of Internet users connected simultaneously and f ( x ) the loading time ex-pressed in seconds. 1 By graphical reading, estimate the loading time, in sec-onds, for 8 000 connected people. 2 a Graphically determine an antecedent of 15 by f . b Give an interpretation of this result. 7. Images, background, definition set: representative curves E.11077 Consider the functions f , g , h , and j , whose representative curves are given below : Give the domain of each of these functions. E.9453 In the coordinate plane, consider the function f whose graph is shown below : 1 Give the domain of the function f . 2 Determine the image of the number 1 by the function f . Justify your approach. 3 Determine the antecedents of the number 1 ; 25 by the function f . Justify your approach. https://chingmath.fr chapExoCorrec/5028 sacados/5028 x-4-3-2-101234y-2-112Cf chapExoCorrec/7042 sacados/7042 0123456789105101520253035404550556065707580859095100105110 chapExoCorrec/11077 sacados/11077 xxyy-3-2-112345-4-3-2-1xxyy-11234567-3-2-11xxyy-5-4-3-2-1123-2-112xxyy123456781234ChCjCfCg chapExoCorrec/9453 sacados/9453 xxyy-4-3-2-10123412Cf
E.9333 Consider the function f defined for all numbers between − 4 and 4 , whose representative curve C f is given in the graph below : 1 Give the domain of the function f . 2 a Determine, justifying your approach, the image of 0 ; 5 by the function f . b Determine, justifying your approach, the set of an-tecedents of − 1 by the function f . 3 Without justifying your answer: a What is the image of the number − 1 by the function f ? b What is the set of antecedents of 2 by the function f ? E.386 On the graph with a coordinate sys-tem, we represent the curve C representing the function f defined for all numbers between − 4 and 4 . 1 Give the domain of the function f . 2 Give the values, justifying your approach : a f ( − 3) b f − 1 2 c f 1 2 d f (0) 3 Give, justifying your approach, the set of antecedents of the following numbers by the function f : a 3 b − 1 c − 2 E.1786 The graph below shows the repre-sentative curve of the function f : 1 Give the domain of the function f . 2 Using the graph, determine the images of the following numbers by the function f : a − 3 b 0 c 2 d 3 3 Using the graph, determine the antecedents of the num-bers below by the function f : a − 1 b 1 4 Which of the following statements are true? a The image of 1 ; 5 by the function f is 2 ; 5 . b 0 ; 5 has only one antecedent by the function f . c By the function f , − 2 ; 5 has no antecedent. 8. Images and background: table of values E.11383 Here are excerpts from a computer program that takes an input value and returns a return value : https://chingmath.fr chapExoCorrec/9333 sacados/9333 x-4-3-2-101234y-2-11234Cf chapExoCorrec/386 sacados/386 x-4-3-2-101234y-2-1123C chapExoCorrec/1786 sacados/1786 x-4-3-2-101234y-3-2-1123Cf chapExoCorrec/11383 sacados/11383 Valeursd’entréesValeursde sorties23691214910f
We denote f as the function associated with this algorithm. 1 Give the value of the sum : f (2) + f (6) + f (9) 2 a Give the set of antecedents of 4 by the function f b Give the set of antecedents of 10 by the function f 9. Algebraic expression and representative curve E.6564 Consider the function f whose expression is defined by the relation: f ( x )=2 x 2 − 3 x +2 Which of the points below belong to the curve C f representing the function f : A (1 ; 2) ; B (4 ; 22) ; C ( − 1 ; 9) ; D (0 ; 3) Hint: answers must be justified. E.9454 Consider the function f defined for any number x by: f ( x ) = 3 x 2 + 4 x − 1 and we note C f the representative curve of this function in the plane provided with a reference frame. Which of the points below belongs to the curve C f : ( − 2 ; − 21) ; ( − 1 ; − 2) ; (0 ; 6) ; (1 ; 12) ; (2 ; 19) Answers will be justified. E.11470 Consider the function f whose ex-pression is defined by the relation: f ( x )=3 x 2 − 2 x +4 Which of the points below belong to the curve C f representing the function f : A (1 ; 5) ; B (2 ; 12) ; C ( − 1 ; 3) ; D ( − 3 ; 37) Hint: answers must be justified. E.11395 Consider the function f defined for all numbers x by: f ( x ) = 0 ; 5 x 2 − 0 ; 5 x − 3 and let C f be the curve representing this function in the plane equipped with a coordinate system. Which of the points below belong to the curve C f : ( − 4 ; − 7) ; ( − 1 ; − 1) ; (0 ; − 3) ; (3 ; 1) ; (4 ; 3) Justify your answers. E.8042 Consider the function f whose ex-pression is defined by the relation: f ( x )= 3 x 2 x − 3 and let’s note C f its representative curve in a reference frame. Which of the points below belong to the curve C f representa-tive of the function f . A (2 ; 2) ; B (0.5 ; − 0.75) Justify your answers. E.8041 Consider the function f whose ex-pression is defined by the relation: f ( x )= x 2 x +4 and let’s note C f its representative curve in a reference frame. Which of the points below belong to the curve C f representa-tive of the function f . A (0 ; 1) ; B 1.5 ; 3 14 Justify your answers. E.11340 Proposition: In a coordinate plane, consider the curve C f representing the function f . The point A ( x A ; y A ) be-longs to the curve C f if the following two conditions are satisfied : x A belongs to the domain of f ( x A ∈D f ) The image of x A is y A ( f x A = y A ) In the plane equipped with a coordinate system O ; I ; J , consider the curve C representing a function f : Which of the expressions below is the expression of the func-tion f : a f ( x ) = 2 x 2 + 3 x − 2 b f ( x ) = 2 x 2 + 4 x + 1 c f ( x ) = 0 ; 25 x 2 + x + 1 d f ( x ) = x 2 + x + 1 https://chingmath.fr chapExoCorrec/6564 sacados/6564 chapExoCorrec/9454 sacados/9454 chapExoCorrec/11470 sacados/11470 chapExoCorrec/11395 sacados/11395 chapExoCorrec/8042 sacados/8042 chapExoCorrec/8041 sacados/8041 chapExoCorrec/11340 sacados/11340 -6-5-4-3-2-12I-12JOCf
E. 11394 In the plane with coordinate sys-tem O ; I ; J , consider the curve C representing a function f : Which of the expressions below is the expression for the func-tion f : a f ( x )=0 ; 5 x 2 +2 ; 5 x +3 b f ( x )= − 0 ; 25 x 2 +0 ; 5 ∗ x +2 c f ( x )= − 0 ; 5 x 2 − 1 ; 75 x +1 d f ( x )=0 ; 25 x 2 +0 ; 75 x +2 E.390 DefinitionfromLePetitLarousse: A multiple-choice questionnaire is a questionnaire that offers several answers for each question asked, from which the correct answer must be chosen. For each question, check the box associated with the correct answer: 1 Let f be a function satisfying f (4)=2 , we say: a antecedent of 4 is 2 . 2 is a solution to the equation f ( x )=2 . 4 has image 2 by the function f . the curve passes through the point with coordinates (2 ; 4) . 2 The representative curve of the function g passes through the point ( − 1 ; 2) , so : the equation g ( x )= − 1 has 2 as a solution. − 1 is an antecedent of 2 by g . 2 has the image − 1 by g . 2 has no image. 3 Let h be a function. The equation h ( x )= − 1 has solu-tions 3 , 1 5 and 2 then : 3 is the unique antecedent of the number − 1 by the func-tion h . the image of the number − 1 is equal to 2 . the representative curve passes through the point with coordinates 2 ; − 1 . the function h verifies h (3)= 2 . 4 So j a function such that the number 3 had image − 5 : j vérifie j ( − 5)=3 . 3 is an antecedent of the number − 5 par the function j . the curve of j passes through the point of coordinate ( − 5 ; 3) . l equation j ( x )= − 5 n admits no solution. E.8027 Consider the function f defined by the algebraic expression : f ( x ) = 12 4 x 2 + 12 In the reference frame below, are given the curves C 1 , C 2 , C 3 representative of functions : Of the three curves proposed, which is the representative curve of the function f ? E.4062 Consider the three curves shown below : For each of the curves, all their points, with coordinates ( x ; y ) , verify the same equation. Associate the corresponding equation with each curve : y = 1 x ; y = x 2 ; y = x 10. Representative curve: definition set E.4394 Definition: Let f be a function. We call the domain of the function f the largest set of numbers x where the image f ( x ) exists. This set is denoted by D f . Consider the function f defined for all numbers between − 3 ; 5 excluded and 4 included, whose representative curve C f is given in the graph below : 1 Give the domain of the function f . https://chingmath.fr chapExoCorrec/11394 sacados/11394 -6-5-4-3-2-1I-12JOCf chapExoCorrec/390 sacados/390 chapExoCorrec/8027 sacados/8027 x-4-3-2-101234y123C1C2C3 chapExoCorrec/4062 sacados/4062 x-101234y-11234C1 x-101234y-11234C2 x-101234y-11234C3 chapExoCorrec/4394 sacados/4394 x-6-5-4-3-2-10123456y-2-112Cf
2 a To justify that the number 5 has no image under the function f , complete the following sentence : The line with equation ::: =5 does not intersect the curve C f of the function f . We can deduce that the number 5 does not have an image . . . . . . by the function f . b Similarly, justify that the number − 4 does not admit an image by the function f . 3 a To justify that the number 1 ; 5 does not admit an image by the function f , complete the following sen-tence : The line with equation ::: =1 ; 5 does not intersect the curve C f of the function f . We can deduce that the number 1 ; 5 does not admit . . . . . . by the function f . b Similarly, justify that the number − 2 does not admit an antecedent by the function f . E.9334 Consider the function f whose curve C f is shown in the O ; I ; J reference below : Which of the boxes below describes the set of values x belong-ing to the definition set of the function f : a − 5 <x< 3.5 b − 5 x< 3.5 c − 5 x 3.5 d − 5 <x 3.5 E.369 Among the graphs shown below, two of them cannot represent a function. Which ones? a b c d 11. Intervals and definition set E.7107 In a reference frame, consider below the curve C f representative of a function f : 1 The numbers x admitting an image by the function f form a set characterized by one of the boxes below. Which is it? a − 3.25 <x< 3 b − 3.25 x< 3 c − 3.25 x 3 d − 3.25 <x 3 2 Give the definition set of the function f in interval form. E.7981 In a reference frame, consider below the curve C f representative of a function f : 1 a Among the intervals below, what is the largest inter-val on which the function f is defined : − 2 ; 5 ; − 2 ; 5 ; − 2 ; 5 ; − 2 ; 5 b Give the definition set of the function f . 2 a Determine the image of the number 0.75 by the func-tion f . Justify your answer. b Determine the set of antecedents of the number − 0.5 by the function f . Justify your answer. https://chingmath.fr chapExoCorrec/9334 sacados/9334 x-6-5-4-3-2-10123456y-2-112Cf chapExoCorrec/369 sacados/369 -6-3036-6-336 -6-3036-6-336 -6-3036-6-336 -6-3036-6-336 chapExoCorrec/7107 sacados/7107 x-4-3-2-101234y-1123Cf chapExoCorrec/7981 sacados/7981 x-2-1012345y-1123Cf
E.366 Consider the following five functions : f : x ↦−→ 1 2 − x ; g : x ↦−→ 2 x + 1 3 x + 3 ; h : x ↦−→ 1 x 2 + 1 j : x ↦−→ 1 − 2 x ; k : x ↦−→ x + 4 1 A quotient is not defined when its denominator is zero. a Can we calculate the image of 2 by the function f ? b For what value does the function g admit no image? c Is there a value that admits no image by the function h . 2 A square root is not defined for strictly negative values. a Can we calculate the image of 5 by the function j ? b For what values of x does the function k not associate images? E.1787 Consider the three functions f , g , h defined by: f ( x ) = 2 x − 1 ; g ( x ) = x − 1 ; h ( x ) = 1 x + 1 1 Is it possible to determine the image of 2 for each of these functions? Justify. 2 Is it possible to determine the image of − 1 for each of these functions? Justify. 3 Give the definition set of each of these functions. 12. Algebraic expression: antecedents E.8030 Consider the linear function f admit-ting as expression : f ( x ) = 0.8 x + 0.2 1 a Solve the equations : f ( x ) = 2 ; f ( x ) = 3 . b Deduce the antecedents of the numbers 2 and 3 by the function f . 2 In the reference frame below, is given the curve ( d ) rep-resentative ( d ) of the function f : Leave construction features to verify the results of ques-tion 1 . E.9336 Method : let f be a function and b a number. To determine the set of antecedents of the number b by the function f , we determine the set of solutions of the equation : f ( x )= b Consider the linear function g defined by the expression : g ( x ) = 1.2 x + 0.1 Determine the antecedent of the number 2.5 by the function g . E.9455 Consider the function f whose image of any number x is defined by: f ( x )=0.35 · x − 0.6 In the plane below, we note C f the representative curve of the function f : 1 Graphically and without justification, give the set of an-tecedents of the number 0.8 by the function f . 2 Determine the set of antecedents of the number 0.3 by the function f . The result will be given as an irreducible fraction. E.8268 Determine the antecedents of the number 4 by the following functions : a h : x ↦−→ 3 · x − 5 b j : x ↦−→ x 2 https://chingmath.fr chapExoCorrec/366 sacados/366 chapExoCorrec/1787 sacados/1787 chapExoCorrec/8030 sacados/8030 x-3-2-1012345y123(d chapExoCorrec/9336 sacados/9336 chapExoCorrec/9455 sacados/9455 xxyy-3-2-1012345-11Cf chapExoCorrec/8268 sacados/8268
E.9457 Consider the calculation program shown in the diagram below : We denote f the function which, to the number x , associates the value returned by this calculation program. 1 Determine the image of the number 2 by the function f . 2 a Give an expression for the function f . b Give the expanded and simplified expression of the function f . 3 Determine the number or numbers supplied as input to the algorithm so that the value 6 is obtained as output from this algorithm. 13. Images and antecedents: algebraic expressions E.11471 Consider the function f whose im-age of any number x is defined by: f ( x )= − 0 ; 25 · x +0 ; 8 In the plane below, equipped with a coordinate system, we denote C f as the curve representing the function f : 1 Graphically and without justification, give the set of an-tecedents of the number 0 ; 8 by the function f . 2 Determine the image of − 4 by the function f . 3 Determine the set of antecedents of − 1 by the function f . E.11472 Consider the function f whose im-age of any number x is defined by: f ( x )=0 ; 2 x +0 ; 6 In the plane below, equipped with a coordinate system, we denote C f as the curve representing the function f : 1 Graphically, give the image of the number 2 by the func-tion f . 2 Determine the image of 6 by the function f . 3 Determine the set of antecedents of 2 by the function f . E.8037 Consider the following calculation program : We denote f the function that associates any number x with the output value of the calculation program when given the value x . 1 Give the algebraic expression of the function f . 2 Complete the table of values below : x − 2 0 3.2 f ( x ) 3 Determine the antecedent of the number − 4.2 by the function f . https://chingmath.fr chapExoCorrec/9457 sacados/9457 Choisir un nombreSoustraire3Soustraire 1Multiplier les deux nombresPrendre le carréSoustraire les deux nombres chapExoCorrec/11471 sacados/11471 xxyy-3-2-1012345-11Cf chapExoCorrec/11472 sacados/11472 xxyy-3-2-101234512Cf chapExoCorrec/8037 sacados/8037 quandest cliquédemanderChoisissez un nombreet attendremettreNombreàréponsemettreNombreàNombre-1mettreNombreàNombre*0,5direNombre
E.8073 Consider the following calculation program : and the function f which, to a number x , entered in the calculation program associates the number returned by this calculation program. 1 Complete the table of values below : x − 5 1 10 f ( x ) 2 Determine the antecedent of the number 3 by the func-tion f . E.384 1 Each of the sentences below defines a function ; deter-mine the algebraic form of each of these functions : a The function f returns x the double of x . b The function g returns the sum of x and the inverse of x . c The function h takes the square root of the product of 4 by the difference of x by 5 . In the following questions, we use the expressions for the func-tions obtained in question 1 : 2 a What is the image of the number 5 by the function f ? b What is the image of the number 7 by the function g ? 3 a Does the number 0 admit an image by the function g ? b Does the number 3 admit an image by the function h ? E.11469 Consider the following calculation program : Let f be the function that associates the input number with the output number of this algorithm. 1 Give the image of the number 4 by the function f . 2 Determine the set of antecedents of the number 1 by the function f . 14. First modeling E.8043 Consider the figure opposite where the square ABCD , the square AMNP and the rectangle MQRB where M is a point on the segment [ AB ] , R is the midpoint of the seg-ment [ BC ] and CD =5 cm . Note x the length of segment [ AM ] . Let f be the function that associates with the value of x the mesude of the area of the hatched part formed by the quadrilaterals AMNP and BMQR . 1 Determine the image of the number 3 by the function f . 2 Give the expression of the function f as a function of x . E.8044 Consider the figure opposite where the square ABCD , the square AMNP and the triangle MRB where M is a point on the segment [ AB ] , R is the midpoint of the seg-ment [ BC ] and CD =5 cm . Note x the length of segment [ AM ] . Let f be the function that associates with the value of x the value of the part hatched formed by the square AMNP and the triangle BMR . 1 Determine the image of the number 3 by the function f . 2 Give the expression of the function f as a function of x . 15. Algebraic expression: use of the calculator https://chingmath.fr chapExoCorrec/8073 sacados/8073 quandest cliquédemanderChoisissez un nombreet attendremettreNombreàréponsemettreNombreàNombre*0,8mettreNombreàNombre+1,4direNombre chapExoCorrec/384 sacados/384 chapExoCorrec/11469 sacados/11469 Choisir un nombreMultiplier par 2Multiplier par 3Ajouter 5Soustraire 5Multiplier les deux nombresAjouter 1 chapExoCorrec/8043 sacados/8043 ABCDMNPQR5cm chapExoCorrec/8044 sacados/8044 ABCDMNPR5cm
E.8024 In the reference frame below, we have plotted the representative curves ( d ) and C of the func-tions f and g respectively. These two functions are defined by the algebraic expressions : f ( x ) = − 0.5 × x + 1 ; g ( x ) = 0.5 x − 1 2 − 1 The aim of the exercise is to obtain the graphical representa-tion of these two functions using the calculator: 1 We’re going to set the calculator display parameters : a Determine the values of the reals xMin , xMax , yMin and xMax so that the four corners of our display have the co-ordinates : ( xMin ; yMin ) , ( xMax ; yMin ) , ( xMax ; yMax ) , ( xMin ; yMax ) . b Let’s set the calculator display window : Calculators TI The key ˇFenêtreı is used Calculators Casio We use the option V-WINDOW (F3) Complete the data xMin , xMax , yMin , yMax missing then validate your choice. 2 Enter the algebraic expressions for the functions : The key ˇf(x)ı is used We go to mode ˇGraphı 3 The representative curves are plotted : The curves are plotted using the ˇgrapheı button We use the command ˇdrawı (F6) E.8029 Consider the function f defined by the relation: f ( x ) = 3 x x 2 + 1 + 2 In the reference frame below, part of the curve C f has been given. We wish to complete the table of values below in order to construct the missing part of the curve C f . x − 1.5 − 1 − 0.5 0 0.5 1 f ( x ) 1 We will enter the expression of the function to be stud-ied : Calculators TI Pressing the f ( x ) key enters the expression of the function Calculators Casio We go to mode Table and en-ter the expression. 2 a What is the pitch between two graduations of the x-axis? This value will be called Δ Tbl (TI) or Step (Casio) . b We define the parameters of the table of values we wish to obtain : The Def tabl option specifies the first value TblStart in the table, and the Δ TblStart calculation step. With the command SET (F5), we indicate the first value of the table ( Start ) and the last ( End ) and also the step ( 0.5 ) . 3 We construct the table of values : https://chingmath.fr chapExoCorrec/8024 sacados/8024 (xMin;yMin)(xMax;yMin)(xMax;yMax)(xMin;yMax)x-4-3-2-101234y-1123C(d chapExoCorrec/8029 sacados/8029 x-4-3-2-101234y-11234Cf
We use the option table (above the key graphe ) . We use the option TABL (F6) Complete the table of values for the function f . E.8062 Consider the function f defined by the relation: f ( x ) = 3 x 2 + 1 − 1 2 In the reference frame below, part of the curve C f has been given. 1 Using a calculator, complete the table of values below, rounding the images to the nearest tenth. x − 1.5 − 1 − 0.5 0 0.5 1 f ( x ) 2 Use the table of values above to complete the represen-tative curve of the function f . E.9452 Hint: the following questions will be answered using the calculator. 1 a Plot the representative curve of the square function f defined by: f ( x )= x 2 b Give the antecedents of the number 2 by the function f rounded to the nearest hundredth. 2 Consider the two functions g and h defined by: g ( x ) = 4 − x 2 ; h ( x ) = x + 1 Determine the coordinates of the intersection points of the curves C g and C h rounded to the nearest tenth. 16. Functions defined by pieces E.385 The plane is given the reference frame below. The curve C f is the graphical representation of the function f : 1 Determine graphically the images by the function f of the numbers : − 2 ; 2 2 Justify that the numbers − 1.5 and 5.5 do not admit an image by the function f . 3 Give the definition set of the function f as a union of intervals. E.9350 Let f be a function whose graphical representation is given in the orthonormal frame below : 1 Give the definition set of the function f . 2 a Determine the image of the number 0 by the func-tion f . b Determine the set of antecedents of − 1 by the function f . https://chingmath.fr chapExoCorrec/8062 sacados/8062 x-4-3-2-101234y-11234Cf chapExoCorrec/9452 sacados/9452 chapExoCorrec/385 sacados/385 x-6-5-4-3-2-1012345y-2-112CfCf chapExoCorrec/9350 sacados/9350 x-6-5-4-3-2-1012y-2-1123CfCf
E.9456 Consider the function f whose image of a number x is defined by: f ( x ) = 0.45 × x 2 − x − 2 − 1.5 In the plane below fitted with a reference frame, we give the curve C f representative of the function f : 1 Graphically and without justification, give the set of def-inition of the function f . 2 Without help from the graphical representation, justify that the number 1 does not admit an image by the func- tion f . E.376 Three curves representing functions are shown below. Determine graphically for each of them its set of definition : a b c d 17. Solving equations E.2756 Below is given the representative curve of the function f in a reference frame : Graphically, solve the equations : a f ( x ) = 1.5 b f ( x ) = 4 E.1799 Consider the function f whose rep-resentative curve C f is given below : Graphically and with the precision given by the grid, solve the following equations : a f ( x )= − 1 b f ( x )=1 18. Point of intersection E.8035 A company manufactures metal parts for the automotive industry every day. Daily production varies between 0 and 25 parts. The amount of expense corresponding to the manufacture of x pieces, expressed in euros, is modeled by the function C defined on the interval 0 ; 25 by: C ( x ) = x 3 − 30 x 2 + 400 x + 100 It is assumed that the company sells its daily production ev-ery day. Each piece is sold at a price of 247 euros. Sales are modeled by the function A defined on the interval 0 ; 25 by: A ( x ) = 247 x In the reference frame below are represented the curves C C and C A respectively of the functions C and A : https://chingmath.fr chapExoCorrec/9456 sacados/9456 xxyy-6-5-4-3-2-101234567-2-112Cf chapExoCorrec/376 sacados/376 -6-4-20246-6-4-22 -6-4-20246-4-224 -6-4-20246-4-224 -6-4-20246-4-224 chapExoCorrec/2756 sacados/2756 x-1012345678y-11234 chapExoCorrec/1799 sacados/1799 x-6-5-4-3-2-101234y-2-112Cf chapExoCorrec/8035 sacados/8035 Nombre de pièces par jour024681012141618202224Montant en euros1000200030004000500060007000CCCA
1 Graphically, give the approximate values of the solutions of the equation C ( x )= A ( x ) . 2 What are the times for the company when the expression C ( x ) − A ( x ) is zero? E.8034 Consider the two functions f and g defined on R whose presentations, C f and C g , are given in the orthogonal frame O ; I ; J below : Grahically, determine the set of solutions to the equation : f ( x )= g ( x ) E.8064 The company BBE (Bio Bois Énergie) manufactures and sells wood pellets to fuel boilers and stoves in private homes and communities. The company produces between 1 and 15 tons of pellets per day. Daily manufacturing costs are modeled by the function C defined on the interval 1 ; 15 où x denotes the quan-tity of pellets in tons and C ( x ) the corresponding daily manufacturing cost in hundreds of euros. In the company BBE the selling price of one ton of wood pellets is 300 euros. The company’s daily revenue is therefore given by the function R defined on the interval 1 ; 15 où x denotes the quantity of pellets in tons and R ( x ) the correspond-ing daily revenue in hundreds of euros. On the graph below, we give C and Δ the respective graph-ical representations of the functions C and R in a frame of reference with origin O . The following questions will be answered using the graph, and with the precision permitted by it. No justification is required. 1 a What is the daily revenue when BBE sells 8 tons of pellets? b How many tons of pellets must BBE sell for the daily production cost to be 2 000 e . 2 a Give the solutions of the equation C ( x )= R ( x ) . b Specify the conditions that allow BBE to make a zero profit. 19. Problems E.8065 Consider the figure opposite where ABCD is a square, AMND and MQRB are two rectangles where M and N belong to the segments [ AB ] and [ CD ] respectively, R is the mid-dle of segment [ BC ] and CD =5 cm . We note x the length of segment [ AM ] in centimeters. Let f be the function that associates the area of the hatched part with the length x . 1 Give the definition set of the function f . 2 Give the image of the number 2 by the function f . 3 Determine the value of x so that the area of the hatched part is equal to 16 cm 2 . E.8109 Consider the rectangle below where AB =3 cm and AD =5 cm , the points E and F belong respectively to the segments [ DC ] and [ BC ] such that CE =2 cm . Parallel to the sides of the ABCD rectangle, we construct two segments to highlight the two hatched rectangles above. We note x the measure of segment [ CF ] and consider the function f which associates with x the measure of the area formed by the two hatched rectangles. 1 Give the definition set of the function f . 2 Give the image of the number 2 by the function f . 3 a Give the algebraic expression of the function f . b Deduce the position of the point F so that the area of the hatched rectangles has measure 7.2 cm 2 . https://chingmath.fr chapExoCorrec/8034 sacados/8034 x-5-4-3-2-1012y-224CfCg chapExoCorrec/8064 sacados/8064 0123456789101112131415481216202428323640444852C chapExoCorrec/8065 sacados/8065 ABCDMNQR5cm chapExoCorrec/8109 sacados/8109 ABCDEF5cmx2cm3cm
E.9445 Consider a rectangle ABCD with dimen-sions CD =6 cm and AD =2 cm and point E positioned such that B is the midpoint of segment [ CE ] : Let M be a point on the segment [ AB ] . The polygon BCDME consists of a triangle EBM and a trapezoid BCDM . Let x be the length of segment [ BM ] in centimeters and f the function that associates the value x with the area of polygon BCDME . 1 Give the domain of the function f . 2 a As a function of x , give the expression for the area of the triangle BEM . b Given x , give the expression for the area of the trape-zoid BCDM . c Give the expression for the function f given x . 3 a Solve the equation : f ( x ) = 10 b Give an interpretation of the results of the previous question. Reminder : E.9458 Consider a rectangle ABCD with dimensions CD =6 cm and AD =4 cm . The points E , F , G , H are such that : the quadrilateral CEFG is a square. the quadrilateral CBHG is a rectangle. Let f be the function that associates the length of x with the area of the shaded part. 1 Give the domain of the function f . 2 a Given x , give the expression for the area of triangle CEF . b As a function of x , give the expression for the area of the trapezoid ADFH . c Give the expanded and reduced expression of the func-tion f as a function of x . 3 a Solve the equation : f ( x ) = 7 ; 8 b Give an interpretation of the results of the previous question. Reminder : https://chingmath.fr chapExoCorrec/9445 sacados/9445 ABCDEM2cm6cm petite base(bgrande Base(Bhauteur(hTrapèzeABb×h2base(bhauteur(hTriangleAb×h2 chapExoCorrec/9458 sacados/9458 A EFFACER ... Pas niveau 2nd ABCDEFGH6cmxx4cm petite base (b)grande Base (B)hauteur(h)TrapezeA=B+b§h2base (b)hauteur(h)TriangleA=b§h2
E.8031 A company wants to in-stall a garden on either side of the entrance road to its warehouse. The garden is shown as a dotted line in the represen-tation opposite. The diagram opposite shows the dimensions of the entrance to the hangar. The quadrilateral ABCD is a rectangle. 1 To which interval do the values of x belong? 2 a Express the area A of the grass as a function of x . b Determine the width of the entrance ( PQ ) so that the turf area is 600 m 2 . 3 Below is given the curve representation of the function A giving the area of the lawn as a function of x : The following questions will be answered by graphical reading. Useful construction lines are left dotted line. a What is the area of the lawn when the inlet measures 25 m . b What is the width of the driveway so that the lawn area measures 500 m 2 . E.8071 Paul wants to build a garage at the bottom of his garden. In the diagram below, the hatched area represents the garage positioned on the property line. The plan of the garage is shown below : The lengths indicated ( 1.6 m and 3 m ) are imposed ; the length indicated by the letter x is variable. Consider the function f which, to the length x , associates the area of the garage. 1 Give the definition set of the function f . 2 Determine the image of the number 2 by the function f . 3 a Determine the antecedent of the number 20 by the function f . b Interpret, in a sentence, the results of the previous question. E.9481 The figure below, composed of rect-angle ABCD and square BEFG such that : AD =6 cm ; AB =2 x ; BE = x We place point H on segment [ AD ] such that : DH = x . Let f be the function that, for the length of x , associates the area of the shaded part composed of the trapezoid ABCH and the square BEFG . 1 Determine the expression of the function f as a function of x . 2 Determine the antecedent of the number 33 using the function f . What does the value obtained represent? Reminder : 20. Unclassified exercises https://chingmath.fr chapExoCorrec/8031 sacados/8031 ABCDMNPQ8m40m30mx x01020304050y20040060080010001200CA chapExoCorrec/8071 sacados/8071 8m1;6mx3mLimitedelapropriétéLimitedelapropriété chapExoCorrec/9481 sacados/9481 ABCDEFGHx6 petite base (b)grande Base (B)hauteur(h)TrapezeA=B+b§h2base (b)hauteur(h)TriangleA=b§h2
E.361 Consider the following three functions f : x ↦−→ 3 x + 2 ; g : x ↦−→ 3 x − 1 x + 3 ; h : x ↦−→ x − 5 1 Determine the image of 5 for each of these functions. 2 Determine the antecedents of the number 4 for each of these three functions. 3 For each function, specify whether it is defined for any real number. If not, cite at least one number that admits no image by this function. E.730 1 Consider the three functions f , g , h defined by: f : x ↦−→ x 2 − 3 x + 2 2 x ; g : x ↦−→ 2 x h : x ↦−→ x + 7 x − 3 ; j : x ↦−→ (6 x − 3) 2 − 36 x 2 + 36 x − 9 Determine the images of the number 4 respectively by the functions f , g , h and j . 2 Consider the two functions k , ‘ defined by: k : x ↦−→ 4 x − 5 ; ‘ : x ↦−→ 9 x 2 − 6 x a Determine the set of antecedents of the number 1 2 by the function k . b Determine the set of antecedents of the number − 1 by the function ‘ . (we’ll think of it as factoring) . E.374 Three functions are defined by their algebraic expression : f : x ↦−→ x 2 − 3 x +1 ; g : x ↦−→ 3 x x +1 +1 ; h : x ↦−→ 1 − x 1 Give the definition set of each of these functions. 2 Give, if possible, the images of 0 and 2 by each of these functions. 3 Determine the set of antecedents : a of the number 1 by the function f ; b of the number 5 2 by the function g ; c of the number 9 by the function h . E.2696 1 Below are three functions whose expression has been en-tered on a calculator: a b c Rewrite on your copy these three functions with the usual presentation of mathematical expressions. 2 For each of the functions below, write the characters you would type into a calculator to insert : a f : x ↦→ 1 + 3 + x x 2 − 3 x b f : x ↦→ (1 − 2 x ) × 3 x − 1) c f : x ↦→ x + 1 x + 1 E.6543 1 Consider the functions f , g , h , j , k defined by the rela-tionships : f ( x ) = 3 · x + 1 ; g ( x ) = x 2 − 2 · x + 3 ; h ( x ) = 9 − 8 · x j ( x ) = 6 − 3 · x − 1 + x 2 ; k ( x ) = x 2 − 9 2 For three of these functions, the number − 2 had the num-bers 4 , 5 , 11 as its image, respectively. Without justification, associate the corresponding func-tion with each of these images. 2 Consider the following three functions : ‘ ( x ) = 2 − 3 · x ; m ( x ) = 3 − 2 · x 1 + 2 x ; n ( x ) = 12 − x 2 Determine the antecedents of the number 3 by the func-tions ‘ , m and n . E.364 Consider the function f defined, for any strictly positive real, by: f ( x ) = − x 2 3 + 2 x 1 Give the definition set of the function f . 2 Give, in simplified form, the images of the following num-bers by the function f : a − 2 b 1 c 2 3 Justify that the number 2 is an antecedent of − 1 3 by the function f . E.2743 1 Let f be the function whose image of the number x is defined by the relation: f ( x ) = 3 2 − 5 x a Determine the definition set of the function f . b Determine, in simplified form, the images of the num-bers : − 7 5 ; − 2 2 Let g be the function defined by the following relation: g : x ↦−→ 1 3 − x 2 a Determine the definition set of the function g . b Determine the set of antecedents of the number 1 4 by the function g . E.1801 1 Consider the function f whose image of the number x is defined by: f ( x ) = 1 − x × 2 x + 3 a Determine the definition set of the function f . b Determine, in simplified form, the images of − 1 and 1 2 by the function f . 2 Consider the function g defined by: g ( x ) = 2 x + 1 3 x − 1 a Determine the definition set of the function g . b Determine the image of the number 3 by the function g . https://chingmath.fr chapExoCorrec/361 sacados/361 chapExoCorrec/730 sacados/730 chapExoCorrec/374 sacados/374 chapExoCorrec/2696 sacados/2696 chapExoCorrec/6543 sacados/6543 chapExoCorrec/364 sacados/364 chapExoCorrec/2743 sacados/2743 chapExoCorrec/1801 sacados/1801