Grade 10 / Reference functions 46 exercises (100% corrected)

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-4-3-2-1234I246810121416JO r513-1 r513-3 01234481216C1 -2-101232468C2 ChingQuizz : 4 exercises available for Quizz assessment : 1. Square function: variations E.8285 We will study the square function h defined by: f : x ↦−→ x 2 1 Give the definition set D f of the square function. 2 a Complete the table of values below : x − 3 − 1 − 0.5 0 0.5 2 4 f ( x ) b Draw the representative curve C f in the reference frame below : 3 a For a and b not both zero, establish the following equality: f ( b ) − f ( a ) = b + a b − a . b Deduce the direction of variation of the square root function on R − and on R + . 4 Does the representative curve of the function f have an axis of symmetry or a center of symmetry? E.4853 Let f be the square function : 1 Complete the following table of values : x − 10 −√ 5 − 3 2 0 1 3 2 1+ 2 f ( x ) 2 a What can be said about the comparison of two neg-ative numbers and their squares? b What can be said about comparing two positive num-bers and their squares? E.4865 Proposal: Two positive numbers and their squares are compared in the same order. Two negative numbers and their squares are compared in the opposite order. Compare the following pairs of numbers : a 3 3 ; 5 b − 1 2 5 ; − 1 5 c 2 √ 3 3 ; 3 4 4 d 2 + 5 ; 3 2. Square function: interval images E.4817 Definition: we call image of the interval [ a ; b ] by the function f , the set consisting of the images of all the ele-ments of the interval [ a ; b ] . Example: Here are two animations to get a bet-ter grasp of the interval image. The two graphs below show two parts C 1 and C 2 of the rep-resentative curve of the square function : 1 a Determine the set of numbers formed by all the ab-scissas of the points on the curve C 1 . b Determine the set of numbers formed by all the ordi-nates of the points on the curve C 1 . c Deduce the image of the interval 2 ; 4 . 2 a Determine the set of numbers formed by all the ab- https://chingmath.fr chapExoCorrec/8285 sacados/8285 -4-3-2-1234I246810121416JO chapExoCorrec/4853 sacados/4853 chapExoCorrec/4865 sacados/4865 chapExoCorrec/4817 sacados/4817 r513-1 r513-3 01234481216C1 -2-101232468C2
-2-12I234JOCf -4-3-2-1234I246810121416JO -3-2-123I23456789JOCf -2-12I234JOCf a-abIJO scissas of the points on the curve C 2 . b Determine the set of numbers formed by all the ordi-nates of the points on the curve C 2 . c Deduce the image of the interval − 1 ; 2 . E.416 In the plane provided with a ref-erence frame O ; I ; J orthonor-mal, consider the curve C f repre-sentative of the square function. Graphically and without justifi-cation, give the image of each of the intervals below by the func-tion f : a [2 ; 5 b − 3 ; − 1 c 2.1 ; 3 d − 4 ; − 1.5 E.9365 Below, in a reference frame, is the representative curve of the square function : 1 a Give the two intervals I and J such that : I ⊂ R − ; J ⊂ R + ; I ∪ J = − 1 ; 5 b Deduce the image of the interval − 1 ; 5 by the func-tion f . 2 Without justification, give the image of the following intervals by the square function : a − 4 ; 2 b − 1 ; 3 E.2692 Give, without justification, the im-ages of the following intervals by the square function : a 2 ; 3 b − 5 ; − 1 c − 2 ; 4 E.2831 Give, without justification, the im-age of the following intervals by the square function : a √ 3 ; 5 b − 1 2 ; − 3 7 c − √ 10 2 ; 7 3 E.2081 In the plane provided with a refer-ence frame O ; I ; J orthornormal, is given below the curve C f representative of the square function : Without justification, copy and complete the following assertions : 1 Si x ∈ 1 ; 3 alors x 2 ∈ : : : : : : 2 Si x ∈ − 1 ; 2 alors x 2 ∈ : : : : : : 3 Si x ∈ − 3 ; − 2 ∪ 2 ; 3 alors x 2 ∈ : : : : : : 4 Si x ∈ −√ 11 ; − 2 ∪ √ 2 ; √ 7 alors x 2 ∈ : : : : : : 3. Square function: equations and inequations E.9367 Solve the equations : a x 2 = 2 b x 2 = 0 c x 2 = − 1 E.9361 Consider the square function denoted f whose representa-tive curve C f in a reference frame O ; I ; J is given be-low : Graphically and without jus-tification, give the set of solu-tions of the following inequa-tions : a f ( x ) 1 b f ( x ) 2 c f ( x ) < 3 d f ( x ) > 4 E.9375 Let f be the square function. With- out justification, give the solution sets, in simplified form, of the inequations : a f ( x ) 10 16 b f ( x ) 9 4 c f ( x ) > ı Note: we can rely on a graphical resolution. E.9374 For each question, give the largest set of numbers verifying the inequality or framing : a x 2 < 3 b x 2 1 c 2 x 2 4 Note: we will rely on calculations and graphical resolution with the help of the curve of the square function : https://chingmath.fr chapExoCorrec/416 sacados/416 -2-12I234JOCf chapExoCorrec/9365 sacados/9365 -4-3-2-1234I246810121416JO chapExoCorrec/2692 sacados/2692 chapExoCorrec/2831 sacados/2831 chapExoCorrec/2081 sacados/2081 -3-2-123I23456789JOCf chapExoCorrec/9367 sacados/9367 chapExoCorrec/9361 sacados/9361 -2-12I234JOCf chapExoCorrec/9375 sacados/9375 chapExoCorrec/9374 sacados/9374 a-abIJO
-4-3-2-1234I-4-3-2-1234JO -4-3-2-1234I-4-3-2-1234JOCf 4. Inverse function: variations E.8284 We will study the inverse function f defined by: f : x ↦−→ 1 x 1 Give the definition set D f of the inverse function. 2 a Complete the table of values below : x − 4 − 2 − 1 − 0.5 0.25 1 2 4 f ( x ) b Draw the representative curve C f in the reference frame below : 3 a For a and b not both zero, establish the following equality: f ( b ) − f ( a ) = a − b a · b . b Deduce the direction of variation of the inverse func-tion on R ∗ + . 4 Does the representative curve of the function f have an axis of symmetry or a center of symmetry? E.4854 1 Let f be the function which, to any non-zero number, returns its inverse. Complete the table with the decimal values of the images rounded to the nearest hundredth : x − 10 − 3 − 2 − 0.5 0.2 0.75 2 f ( x ) 2 a What can be said about the comparison of two neg-ative numbers and their inverses? b What can be said about the comparison of two positive numbers and their inverses? E.340 Let a and b be two strictly positive numbers. 1 For a = b , express the difference below as a quotient : 1 a − 1 b . 2 Suppose that b<a : a Determine the sign of : 1 a − 1 b . b Compare : 1 a and 1 b . E.9381 Compare the following pairs of num-bers : a 3 ; 6 b 1 ı ; 1 3 c − 1 5.15 ; − 1 5.105 5. Inverse function: interval images E.2818 Consider the inverse function de-noted f whose representative curve C f in a reference frame O ; I ; J is given below : https://chingmath.fr chapExoCorrec/8284 sacados/8284 -4-3-2-1234I-4-3-2-1234JO chapExoCorrec/4854 sacados/4854 chapExoCorrec/340 sacados/340 chapExoCorrec/9381 sacados/9381 chapExoCorrec/2818 sacados/2818 -4-3-2-1234I-4-3-2-1234JOCf
-4-3-2-1234I-4-3-2-1234JO -4-3-2-1234I-4-3-2-1234JOCf -4-3-2-1234I-4-3-2-1234JOCf -3-2-123I-3-2-123JO −∞0∞∞0−∞0xVariationdef Graphically and without justification, give the image by the function f of the following intervals : a 1 ; 2 b − 3 ; − 0.5 c 0.25 ; 4 E.9363 Give, without justification, the im-age of the following intervals by the inverse function : a − 4 ; − 1 b 2 ; 5 2 c 0.0001 ; 1.5 E.9373 Without justification, give the image of the intervals below by the inverse function : a 10 − 1 ; 10 5 b 10 − 10 ; 10 − 9 c − 10 6 ; − 10 5 E.2161 Consider the following representa-tions of the square and inverse functions in the marks below : Using the graphical representa-tions above, determine the im-ages of the following sets by the inverse function : a − 3 ; − 1 b 1 9 ; 1 2 c − 1 ; 0 ∪ 0 ; 1 E.9366 In an orthonormal coordinate sys-tem O ; I ; J , the curve C f representing the inverse function is given : Complete the following state-ments without justification : 1 If x ∈ 2 ; 4 then 1 x ∈ :::::: 2 If x ∈ 0 ; 2 then 1 x ∈ :::::: 3 If x ∈ −∞ ; − 1 2 then 1 x ∈ :::::: E.9364 Give, without justification, the im-ages of the following intervals by the inverse function : a 1 ; 2 b 0 ; 5 c 3 ; + ∞ E.9376 Consider the inverse function de-noted f . Without justification, give the images of the intervals below by the function f : a 3 ; + ∞ b −∞ ; − 1 c 0 ; 1 2 6. Inverse function: equation and inequation E.9368 Solve the following equations : a 1 x = 2 b 1 x = 0 c 1 x = − 4 E.9362 Consider the inverse function noted f whose representative curve C f in a reference frame O ; I ; J is given below : Graphically and without justifi-cation, give the set of solutions of the following equations and inequations : a 1 x 1 b 1 x < − 3 c 1 x > 7 E.9377 Representation of the inverse function : Let f be the inverse function. Below are the representative curve and the table of variations for the function f : Based on the representative curve or the table of variations of the inverse function, give, without justification, the sets of solutions for each of the following inequalities: a 1 x 1 5 b 1 x > 3 4 c 1 x < − 1 3 E.9378 Without justification, give the set of solutions of the following inequalities: a 1 x > − 1 b 1 x < 2 7. Square root: variation E.403 We will study the square root func-tion h defined by: f : x ↦−→ x https://chingmath.fr chapExoCorrec/9363 sacados/9363 chapExoCorrec/9373 sacados/9373 chapExoCorrec/2161 sacados/2161 -4-3-2-1234I-4-3-2-1234JO chapExoCorrec/9366 sacados/9366 -4-3-2-1234I-4-3-2-1234JOCf chapExoCorrec/9364 sacados/9364 chapExoCorrec/9376 sacados/9376 chapExoCorrec/9368 sacados/9368 chapExoCorrec/9362 sacados/9362 -4-3-2-1234I-4-3-2-1234JOCf chapExoCorrec/9377 sacados/9377 -3-2-123I-3-2-123JO −∞0∞∞0−∞0xVariationdef chapExoCorrec/9378 sacados/9378 chapExoCorrec/403 sacados/403
-123456789I-1234JO xy0369123 0∞0∞xVariationdef 1 Give the definition set D f of the square root function. 2 a Complete the table of values below : x 0 0.125 0.25 1 2 4 9 f ( x ) b Draw the representative curve C f in the frame below : 3 a For a and b not both zero, establish the following equality: a − b = a − b a + b . b Deduce the direction of variation of the square root function on R + . 4 Does the representative curve of the function f have an axis of symmetry or a center of symmetry? 8. Square root: interval image E.4515 Representation of the square root function : Below are the representative curve and the table of variation of the square root function : Without justification, give the images of the intervals below by the square root function : a 1 ; 4 b 0.81 ; 2.25 c 5 ; 9 9. Square root: equation and inequation E.4972 Without justification, give the solu-tion set of the following inequalities: a x> 4 b x< 9 E.4535 Without justification, give the set of solutions of the following inequalities: a x> 3 b x< 5 10. Cube function: variations E.8286 We will study the cube function h defined by: f : x ↦−→ x 3 1 Give the definition set D f of the cube function. 2 a Complete the table of values below : x − 2 − 1 − 0.5 0 0.5 1 2 f ( x ) b Draw the representative curve C f in the reference frame below : https://chingmath.fr -123456789I-1234JO chapExoCorrec/4515 sacados/4515 xy0369123 0∞0∞xVariationdef chapExoCorrec/4972 sacados/4972 chapExoCorrec/4535 sacados/4535 chapExoCorrec/8286 sacados/8286
-2-12I-8-6-4-22468JO xy-2-1012-6-4-2246 −∞∞−∞∞xVariationdef 3 a For a and b not both zero, establish the following equality: f ( b ) − f ( a )= b − a b 2 + a · b + a 2 b Deduce the direction of variation of the cube function on R − and on R + . 4 Does the representative curve of the function f have an axis of symmetry or a center of symmetry? E.9382 Proposition: two numbers and their cubes are compared in the same order. 1 Let a and b be two strictly negative numbers such that a<b . Compare the two numbers a 3 and b 3 . 2 Let a and b be two strictly negative numbers such that a 2 <b 2 . Compare the two numbers a 3 and b 3 . 11. Cube function: interval images E.9380 Representation of the cube function : Below are the representative curve and the table of variation of the cube function : Without justification, give the images of the intervals below by the cube function : a 1 ; 4 b − 3 ; − 1 c − 3 4 ; − 3 2 12. Cube function: equations and inequations E.4536 Without justification, answer the following questions : 1 Solve the inequation : x 3 > 8 2 Solve the inequation : x 3 27 13. Relative positions of reference functions E.305 1 Complete the following table : a 0 0 ; 3 0 ; 7 1 10 a 2 0 ; 04 18 ; 49 a 3 1 ; 728 2 Let x be a positive real number ( x ∈ R + ) : a Conjecture the values of x that satisfy the equality: x = x 2 ? b Conjecture the values of x that satisfy the inequality: x <x 2 ? c Conjecture the values of x that satisfy the inequality: x > x 2 ? 3 Let E denote the expression : E = x 2 − x . a Factor the expression E . b Examine the sign of E with respect to 0 ; 1 . Use this to determine the comparison of x and x 2 with respect to 0 ; 1 . c Examine the sign of E with respect to 1 ; + ∞ . Use this to determine the comparison of x and x 2 with re-spect to 1 ; + ∞ . 4 Using a method similar to that in the previous questions, compare the numbers x 2 and x 3 for any positive value of x ( x ∈ R + ) . https://chingmath.fr -2-12I-8-6-4-22468JO chapExoCorrec/9382 sacados/9382 chapExoCorrec/9380 sacados/9380 xy-2-1012-6-4-2246 −∞∞−∞∞xVariationdef chapExoCorrec/4536 sacados/4536 chapExoCorrec/305 sacados/305
23I2JOCfCgCh E.2703 Consider the three functions f , g , h defined by: f : x ↦−→ x ; g : x ↦−→ x 2 ; h : x ↦−→ x whose representative curves are given in the reference frame O ; I ; J orthonormal below : 1 Graphically, investigate the relative position of the curves C f , C g and C h on R + . Let’s establish the result of the previous question alge-braically: 2 a Draw up the sign table for the expression x 2 − x . b Compare the functions f and g on each of the intervals 0 ; 1 and 1 ; + ∞ . 3 a For x ∈ 0 ; + ∞ , establish the equality: f ( x ) − h ( x ) = x 2 − x x + √ x b Compare the functions f and h on each of the intervals 0 ; 1 and 1 ; + ∞ . E.2959 1 Consider the two functions f and g defined by: f : x ↦−→ x − 1 2 ; g : x ↦−→ x 2 − 1 2 Compare the functions f and g on the interval 0 ; 1 . 2 Consider the functions h and j defined by: h : x ↦−→ 1 2 √ x + 1 ; j : x ↦−→ 1 2 x + 1 Compare the functions h and j on each of the intervals 0 ; 1 and 1 ; + ∞ . E.9372 Consider the two functions f and g defined on R by the expressions : f ( x ) = x 4 ; g ( x ) = x In the plane provided with a reference frame, note C f and C g the representative curves of the functions f and g . 1 Determine the polynomial P verifying equality: f ( x ) − g ( x ) = x · x − 1 ·P x 4 + x 2 Deduce the relative position of the curves C f and C g on the intervals 0 ; 1 et 1 ; + ∞ . 14. Share E.2729 We denote by f the square function. Say whether the following assertions are ˇ true ı or ˇ fausses ı. In the case where an assertion is false, a counterexample will be cited : 1 If x> 1 then x 2 > 1 . 2 If x 2 > 1 then x> 1 . 3 The image of the interval − 3 ; 4 by the function f is 9 ; 16 . 4 The image of the interval − 5 ; 1 by the function f is 0 ; 25 . 5 − 4 < 1 = ⇒ f ( − 4) <f (1) . 6 The function f is increasing at R . 15. Unclassified exercises E.9379 Without justification, give the im-ages of the intervals below by the square root function : a 10 4 ; 10 20 b 2 16 ; 2 32 https://chingmath.fr chapExoCorrec/2703 sacados/2703 23I2JOCfCgCh chapExoCorrec/2959 sacados/2959 chapExoCorrec/9372 sacados/9372 chapExoCorrec/2729 sacados/2729 chapExoCorrec/9379 sacados/9379
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