Outside the high school program / Associated functions and compound functions 54 exercises (including 44 corrected)

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CfI -6-5-4-3-2-123456I-3-2-123JOCf -3-2-123I-3-2-123JOCg 1. Introduction of compound E.2197 Let f be the defined function whose image of x is defined by: f ( x ) = x − 1 x − 2 1 Establish the following equality: x − 1 x − 2 =1+ 1 x − 2 We denote g the inverse function. In the previous question, we just established that : f ( x ) = g ( x − 2) + 1 2 By what transformation of the plane, is the curve C f rep-resentative of the function f obtained from the hyperbola C g representative of the inverse function? The figure below represents the hyperbola obtained by the inverse function and I its center of symmetry. 3 Correctly place the reference frame so that this curve is the representation of the function f . E.2215 Consider the plane with an orthonormal coordinate system ( O ; I ; J ) . 1 Let f be a function defined on [ − 6 ; 6] whose representa-tive curve is as follows : a Determine the set of antecedents of the number − 1 . b Determine images for the following intervals : − 6 ; − 1 ; 3 ; 6 ; − 3 ; 2 − 5 ; − 3 ∪ 0 ; 1 c Determine the antecedents of all numbers forming the interval − 2 ; − 1 2 Consider also the function g defined on − 3 ; 3 whose graphical representation is as follows : a Determine the images of the following numbers by the function g ◦ f : − 3 ; 4 b Determine the image of the interval 0 ; 4 by the func-tion g ◦ f . 2. Composed of functions E.1159 Consider the function f whose represen-tative curve is given in the reference frame O ; I ; J below : https://chingmath.fr chapExoCorrec/2197 sacados/2197 CfI chapExoCorrec/2215 sacados/2215 -6-5-4-3-2-123456I-3-2-123JOCf -3-2-123I-3-2-123JOCg chapExoCorrec/1159 sacados/1159
-4-3-2-1234I-3-2-123JOCfC1C2C3C4C5 -4-3-2-1234I-3-2-123JOCf -3-2-123I-123456789JOCfCg -3-2-123I-6-5-4-3-2-123456JOC1C2C3C4C5C6 In this frame of reference is also given the representative curves associated with the function f whose algebraic expres-sions are as follows : g : x ↦→ f ( x +2) ; h : x ↦→ f (2 · x ) ; j : x ↦→ f ( x − 2) k : x ↦→ 2 · f ( x ) ; ‘ : x ↦→ f ( x ) − 2 Associate each of these functions with its representative curve. E.2158 Let f be a function defined on the in-terval [ − 4 ; 2] whose representation is given in the reference frame ( O ; I ; J ) below : We define the function g by the relation: g : x ↦−→ f ( x − 2) 1 For what value of x , are you able to give the value of g ( x ) ? What is the defining set of the function g ? 2 Complete the following table of values : x − 4 − 2 0 1 2 3 4 f ( x ) × × g ( x ) × 3 Draw the curve C g representative of the function g . 4 What can you say about the curves C f and C g ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E.2702 Consider the two reference marks ( O ; I ; J ) in which have been drawn representative curves of polynomials of second degree : 1 The curve C f is the representative curve of the function f defined by: f : x ↦−→ x 2 + 2 x + 4 The function g is defined, explicitly by the function f , by the following relation: g ( x ) = f ( x + ¸ ) + ˛ where ¸ ∈ R and ˛ ∈ R . a Using the representations of these two functions, de-termine the values of the real numbers ¸ and ˛ . b Deduce the reduced developed form of the expression of f ( x ) . 2 The curves C 1 to C 6 are representations of functions de-fined by the relation: x ↦−→ ¸ · x 2 where ¸ is a real number. Determine for each function the value of ¸ . https://chingmath.fr -4-3-2-1234I-3-2-123JOCfC1C2C3C4C5 chapExoCorrec/2158 sacados/2158 -4-3-2-1234I-3-2-123JOCf sacados/2702 -3-2-123I-123456789JOCfCg -3-2-123I-6-5-4-3-2-123456JOC1C2C3C4C5C6
−∞08∞-11-3∞xVariationdeg xVariationdef−5−214−21−33 3. Compound and expressions E.2154 In each of the following questions, give the algebraic expression of f ◦ g ( x ) as a function of x : a f ( x ) = 3 x + 1 et g ( x ) = 2 − x b f ( x ) = x 2 et g ( x ) = x − 4 c f ( x ) = x − 4 et g ( x ) = x 2 d f ( x ) = x et g ( x ) = 3 x + 2 e f ( x ) = 3 x + 2 et g ( x ) = 1 x f f ( x ) = − 2 x + 1 et g ( x ) = 2 x + 1 E.2166 Consider the following three functions defined on R : f : x ↦−→ − 1 2 x + 1 ; g : x ↦−→ x 2 ; h : x ↦−→ 2 x − 1 We define a new function F called ˇ the composite of functions f , g and h ı : The value of the image of x is the value of successive images of x by the functions f , g and h . The following diagram shows this situation for the special case x =6 : F : 6 f ↦−→ − 2 g ↦−→ 4 h ↦−→ 7 1 a Show that F (4) = 1 . b Determine the images of − 4 and 3 by the function F . c Determine the set of antecedents of 97 by the function F . 2 a Which of the following three expressions represents F ( x ) : A = − 1 2 · 2 x − 1 2 + 1 ; B = 2 · − 1 2 x + 1 2 − 1 C = − 1 2 x + 1 + x 2 + 2 x − 1 b Give the expanded and reduced form of the function F 3 For each question, give an algebraic writing of the func-tion F obtained by composition of the functions f , g and h (no factoring or development required) . a f : x ↦−→ x 2 ; g : x ↦−→ 3 x + 2 ; h : x ↦−→ 1 x b f : x ↦−→ 1 x ; g : x ↦−→ x ; h : x ↦−→ 3 x + 1 4 Consider the function F obtained by composing the func-tions f , g , h defined below : f : x ↦−→ 2 x ; g : x ↦−→ 3 x + 2 ; h : x ↦−→ x a Perform the following calculation: h g f (2) b What does this calculation represent with respect to the function F ? c Give the algebraic writing of the function F The compound of f , g , h is noted h ◦ g ◦ f E.4652 Consider the function f defined on R + by the relation: f ( x ) = 1 − √ x 2 1 For x ∈ 0 ; 1 , establish the following equality: f ◦ f ( x ) = x 2 For x ∈ 1 ; + ∞ , determine a simplified expression for the function f ◦ f . 4. Compound and variation table E.3308 Consider the function f defined on R by the following relation: f ( x ) = − 2 x 2 − 4 x + 6 1 a Determine the factored form of the expression f ( x ) . b Draw up the table of variations of the function f . (Also indicate the two roots of the function f ) 2 Consider the function g defined on R with the following table of variations : a Make a conjecture about the value of the following lim-its : lim x ↦→ + ∞ g ◦ f ( x ) ; lim x ↦→−∞ f ◦ g ( x ) b Justify that the function f ◦ g is decreasing on the in- terval −∞ ; 0 . c Justify and specify the monotonicity of the composite g ◦ f on each of the following two intervals : −∞ ; − 3 ; − 3 ;1 E.2300 Consider a function f defined on − 5 ; 4 whose table of variations is given below : Determine the table of variations of the functions associated with f shown below : a g : x ↦−→ f ( x +2) b h : x ↦−→ − 2 · f ( x ) c j : x ↦−→ f ( x − 2) + 1 https://chingmath.fr chapExoCorrec/2154 sacados/2154 chapExoCorrec/2166 sacados/2166 sacados/4652 chapExoCorrec/3308 sacados/3308 −∞08∞-11-3∞xVariationdeg chapExoCorrec/2300 sacados/2300 xVariationdef−5−214−21−33
E.1160 Consider the following functions : g : x ↦−→ 1 − 2 x ; h : x ↦−→ x ; j : x ↦−→ x 2 k : x ↦−→ 1 x ; ‘ : x ↦−→ 2 x 2 − 3 x + 1 1 For each question, give the simplified writing of f ( x ) a f = k ◦ g ◦ k b f = g ◦ j ◦ k c f = j ◦ h d f = l ◦ j 2 Consider the function m defined by the following compo-sition string : m : x , −−−−−→ ‘ ( x ) h , −−−−−→ h ‘ ( x ) a Determine the definition set of the function m . b Deduce the table of variations of the function m . Jus-tify your approach. E.2162 Consider the function f , defined on R , whose image of a number x is defined by: f ( x ) = − 3 2 x − 1 2 + 1 Let g be the linear function defined by: g : x ↦−→ 2 x − 1 . 1 Determine the interval I such that : g I = R + 2 Justify that the function f is monotonic on I . 3 Specify the direction of variation of the function f on each of the intervals I and R \ I . E.2164 Give the direction of variation of the functions below on the specified interval. No justification is required : a f : x, −−−−−→ 2 − 3 x on −∞ ; 2 3 b g : x, −−−−−→ 2(3 − x ) 2 +1 on [3 ; + ∞ [ c h : x, −−−−−→ 2 ( x +1) 2 +1 on ] −∞ ; − 1[ d j : x, −−−−−→ x 2 − 3 x +2 on R − e k : x, −−−−−→− 3 x − 1 on R + f ‘ : x, −−−−−→− 2 3 − x +1 on ] −∞ ; 3] E.2195 1 Consider the following two functions : f : x ↦−→ 3 x − 2 ; g : x ↦−→ 5 − x Determine the functions f ◦ g and g ◦ f . 2 Let h be the function defined by: h : x ↦−→ 2 ( x + 1) 2 − 1 a Write the string of compound reference functions char-acterizing h . b Justify and specify the monotonicity of the function f on ] −∞ ; − 1[ . 3 Without any justification, give the direction of variation of the following functions on the specified interval: a j : x ↦−→ − 3( x + 1) 2 on − 1 ; + ∞ b k : x ↦−→ x + x 2 on R + c ‘ : x ↦−→ − 2 x 2 − 1 + 2 on −∞ ; − 1 d m : x ↦−→ 3 x − 1 − ( x − 2) 2 on 1 3 ; 2 E.2217 Consider the function f whose image of a number x is defined by: f ( x ) = ( x + 1)(2 − x ) 1 Determine the definition set of the function f . 2 a Show that for any x ∈D f , we have : f ( x ) = − x − 1 2 2 + 9 4 b Determine the reference function composition string characterizing the f function. c Justify the monotonicity of the function f on the two intervals : I = D f ∩ −∞ ; 1 2 and J = D f ∩ 1 2 ; + ∞ d Draw up the table of variations of the function f . 5. Composed of functions E.421 Most of the functions used are compound functions from reference functions. The function x ↦−→ x 2 +1 can be seen as the compound of reference functions in the following way: x f , −−−−−→ x 2 g , −−−−−→ x 2 + 1 h , −−−−−→ √ x 2 + 1 où f : x ↦−→ x 2 g : x ↦−→ x + 1 h : x ↦−→ x 1 Decompose the following functions using reference func-tions : a x , −−−−−→ 3 x 2 − 1 b x , −−−−−→ 2 3 + x 2 c x , −−−−−→ 2 − 2 x + 3 2 We define a and b two elements of the interval 1 ; 5 such that a<b . For each of the functions in question 1 , compare the images of the numbers a and b . 3 Analyzing your results from question 1 , complete the following two sentences : The compound of two increasing functions is . . . . . . . . . The compound of an increasing and a decreasing func-tion is . . . . . . . . . https://chingmath.fr chapExoCorrec/1160 sacados/1160 chapExoCorrec/2162 sacados/2162 chapExoCorrec/2164 sacados/2164 chapExoCorrec/2195 sacados/2195 chapExoCorrec/2217 sacados/2217 sacados/421
4 What can be said about the direction of variation of the sum of two increasing functions? of two decreasing func-tions? of an increasing function and a decreasing func-tion? Justify each of your statements with a demonstration or counterexample. E.1943 Consider the following three functions : f : x ↦−→ − 1 2 x + 1 ; g : x ↦−→ x 2 ; h : x ↦−→ 2 x − 1 We define a new function F called ˇ the composite of functions f , g and h ı : The image of x has as its value the value of successive images of x by the functions f , g and h . The following diagram represents this situation for the special case x =6 : F : 6 f ↦−→ − 2 g ↦−→ 4 h ↦−→ 7 1 a Show that F (4)=1 . b Determine the images of − 4 and 3 by the function F . c Determine the antecedent of 97 by the function F . 2 a Which of the following three expressions represents F ( x ) : A = − 1 2 (2 x − 1) 2 + 1 ; B = 2 − 1 2 x + 1 2 − 1 C = − 1 2 x + 1 + x 2 + 2 x − 1 b Give the expanded and reduced form of the function F 3 For each question, give an algebraic writing of the func-tion F obtained by composition of the functions f , g and h (no factoring or development required) . a f : x ↦−→ x 2 ; g : x ↦−→ 3 x + 2 ; h : x ↦−→ 1 x b f : x ↦−→ 1 x ; g : x ↦−→ x ; h : x ↦−→ 3 x + 1 4 Consider the function F obtained by composing the func-tions f , g , h defined below : f : x ↦−→ 2 x ; g : x ↦−→ 3 x + 2 ; h : x ↦−→ x a Perform the following calculation: h g f (2) b What does this calculation represent with respect to the function F ? c Give the algebraic writing of the function F E.1944 Consider the function F defined as the composite of the following three reference functions : f : x ↦−→ 2 x − 2 ; g : x ↦−→ x 2 ; h : x ↦−→ − x + 1 1 a Determine the direction of variation of the functions f and h on R . b Give the intervals on which the function g is monotonic. Specify. We wish to study the direction of variation of the function F only on the interval ] −∞ ; 1] . To do this, consider in the rest of the exercise two real num-bers a and b belonging to the interval ] −∞ ; 1] verifying the relationship : a <b : 2 Justify the following inequality: f ( a ) <f ( b ) 3 a What is the direction of variation of the function g on R − ? b What is the sign of the two numbers 2 a − 2 and 2 b − 2 ? c Deduce the comparison of : (2 a − 2) 2 et (2 b − 2) 2 4 a Using the direction of variation of the function h , determine the direction of comparison of the following numbers : − (2 a − 2) 2 +1 et − (2 b − 2) 2 +1 5 Deduce the direction of variation of the function F on ] −∞ ; 1] E.1964 Let f be a function whose image of a number x is defined by: f : x ↦−→ 1 √ x + 1 1 Give the definition set of the function f . 2 Decompose this function using three reference functions. 3 Prove the decay of the function f over D f . E.1966 Consider the function f whose images are algebraically defined by: f : x ↦−→ − 2 x 2 − 1 1 Give the set D f definition of the function f . 2 Decompose function f using four reference functions. 3 a Sur ] −∞ ; 0] ∩D f , establish the decay of the function f b Prove the growth of the function f on the interval [0 ; + ∞ [ ∩D f 4 Give the table of variations of the function f . https://chingmath.fr sacados/1943 sacados/1944 sacados/1964 sacados/1966
E.2124 Consider the function F obtained by com-posing the functions f , g , h following : f : x ↦−→ 2 x + 4 ; g : ↦−→ 1 x h : ↦−→ 1 − x 1 a Give, among the algebraic expressions below, the one that represents the function F : A =1 − 2 · 1 x +4 ; B =(2 x +4)+ 1 x +(1 − x ) ; C =1 − 1 2 x +4 b Justify that : f ( x ) = 2 x + 3 2 x + 1 2 a Give the direction of variation of the functions f and h on the interval R . b Give the intervals on which the function g is monotonic. Specify. 3 To study the direction of variation of the function F on the interval −∞ ; − 2 , we consider two numbers a and b belonging to −∞ ; − 2 and verifying: a <b a Compare f ( a ) and f ( b ) . b What is the sign of 2 a +4 and 2 b +4 ? c Give the direction of variation of the function g on R − . d Deduce the direction of comparison of 1 2 a +4 and 1 2 b +4 . e Compare the following two numbers : 1 − 1 2 a +4 ; 1 − 1 2 b +4 . f Deduce the direction of comparison of F ( a ) and F ( b ) . g Give the direction of variation of the function F on −∞ ; − 2 . E.1761 1 After observing the direction of variation on your cal- culator of each of the following functions, support your observation with an algebraic proof. a Let f be the function defined by the formula : f ( x ) = x + 3 on R . b Let g be the function defined by the formula : g ( x ) = − 1 4 x − 1 on R . c Let j be the function defined by the formula : j ( x ) = 3(1 − x ) 2 + 2 at R . d Let k be the function defined by the formula : k ( x ) = x on the interval R + . e Let l be the function defined by the formula : l ( x ) = 1 x 2 on R ∗ 2 a Establish identity: x 2 +2 x +1=( x +1) 2 b Consider the function m defined on R by the relation: m ( x )= x 2 +2 x +1 . Establish the strict decay of the function m on ] − ∞ ; − 1] . E.1758 1 Study the direction of variation of the following func-tions : f defined on R by the relation: f ( x )=2 x − 1 g defined on R by the relation: g ( x )= − 2 x − 1 h defined on R by the relation: h ( x )= x 2 k defined on R by the relation: k ( x )=( x − 1) 2 − 2 l defined on R ∗ by the relation: l ( x )= 1 x m defined on R \{ 3 } by the relation: m ( x )= 1 ( x − 3) 2 2 a Establish the relationship : x 2 − 2 x − 1=( x − 1) 2 − 2 b Establish the direction of variation of the function n defined on R by the relation: n ( x ) = x 2 − 2 x − 1 6. Operations on functions E.2148 1 For each question, give without justification the direction of variation of the functions f , g and f + g : a f : x ↦−→ 2 x + 1 ; g : x ↦−→ 3 x + 1 b f : x ↦−→ 3 x + 1 ; g : x ↦−→ 3 − 2 x c f : x ↦−→ − 3 x + 1 ; g : x ↦−→ x − 1 2 Consider the two functions f and g defined by: f : x ↦−→ 2 x − 2 ; g : x ↦−→ 3 − x a Expand expression f ( x ) · g ( x ) . b Draw up the table of variations of the function f · g . https://chingmath.fr sacados/2124 sacados/1761 sacados/1758 chapExoCorrec/2148 sacados/2148
-4-3-2-1234I-1234JOCf -4-3-2-1234I-3-2-123JOCf E.2157 Let f be a function defined on the in-terval − 4 ; 4 whose representation is given in the reference frame ( O ; I ; J ) below : Consider the table below : x -4 -2 0 0.5 2 2.5 3 4 f ( x ) g ( x ) 1 Complete the line of values taken by the function f . 2 Let g be the function defined on the interval [ − 4 ; 4] whose image of a number x ∈ [ − 4 ; 4] is given by the rela-tion : g ( x ) = f ( x ) + 2 a Complete the third row of the above table of values. b In the reference frame O ; I ; J , draw the curve C g representative of the function g . 3 What relationship exists between the curves C f and C g ? E.2159 Let f be a function defined on the inter-val [ − 4 ; 4] whose representation is given below in the ( O ; I ; J ) orthonormal frame : Let g and h be two functions defined on [ − 4 ; 4] by: g ( x ) = 2 · f ( x ) ; h ( x ) = 1 2 · f ( x ) 1 Complete the following table of values : x − 4 − 3 − 2 − 5 4 0 5 4 2 3 4 f ( x ) g ( x ) h ( x ) 2 Draw the representative curves C g and C h respectively of the functions g and h in the above reference frame. https://chingmath.fr chapExoCorrec/2157 sacados/2157 -4-3-2-1234I-1234JOCf chapExoCorrec/2159 sacados/2159 -4-3-2-1234I-3-2-123JOCf
-4-3-2-1234I-3-2-123JOCf -∞−214−5−22−1Variationdefx −21∞-∞2−1xVariationdef 137451xVariationdef E.2160 Let f be a function defined on the in-terval [ − 4 ; 4] whose representation is given in the reference frame ( O ; I ; J ) below : Consider the function g defined on [ − 4 ; 4] by the relation: g ( x ) = − f ( x ) 1 Complete the following table of values : x -4 -3 -2 -1 0 1 2 2.5 4 f ( x ) g ( x ) 2 Draw the curve C g representative of the function g . 3 What relationship exists between the curves C f and C g ? E.2695 Consider the two functions f and g de-fined on R : f : x ↦−→ x + 2 ; g : x ↦−→ (3 x − 2)(2 x + 4) 1 Determine the simplified form of the following expres-sions : a f + g ( x ) b f × g ( x ) c f g ( x ) 2 Give the definition set of the function f g . E.408 Consider the function f defined on −∞ ; 4 whose table of variations is given below : Consider the functions g , h and j defined by: g ( x ) = f ( x ) + 2 ; h ( x ) = 2 · f ( x ) ; j ( x ) = − f ( x ) Draw up the tables of variation of the functions g , h and j . E.6572 Consider the function f admitting the following table of variations : Draw up, without justification, the table of variations of the function g defined by the relation: g ( x ) = 3 · f 2 · x E.7378 Consider the function f defined on 1 ; 7 whose table of variations is given below : 1 Without justification, draw up the tables of variations of the functions g and h defined by: g ( x ) = 2 · f ( x ) ; h ( x ) = f ( x ) + 2 2 Consider the function j defined by the relation: j ( x ) = f ( x − 2) a Give the definition set of the function j . b Draw up, without justification, the table of variations of the function j . 7. Composition: roots E.4983 Consider the second-degree polynomial function f defined by the relation: f ( x ) = − x 2 − 2 · x + 3 We construct the function g by the relation: g : x ↦−→ f ( x ) 1 Conjecture about the function g : After plotting the representative curve of the function g using the calculator: a Conjecture the set of definition of the function g . b Conjecture the direction of variation of the function g on the intervals − 3 ; − 1 and − 1 ; 1 . 2 Study of function g : a Determine the zeros of the function f , then complete the table of variations below : https://chingmath.fr chapExoCorrec/2160 sacados/2160 -4-3-2-1234I-3-2-123JOCf chapExoCorrec/2695 sacados/2695 chapExoCorrec/408 sacados/408 -∞−214−5−22−1Variationdefx chapExoCorrec/6572 sacados/6572 −21∞-∞2−1xVariationdef chapExoCorrec/7378 sacados/7378 137451xVariationdef chapExoCorrec/4983 sacados/4983
−∞.........∞−∞0...0−∞xVariationdef −∞.........∞∞0...0∞xVariationdef b Justify that the function g is defined on − 3 ; 1 . c Justify that the function g is decreasing on − 1 ; 1 . E.4982 1 Consider the function f whose image of x is given by the relation: f ( x ) = 2 · x − x 2 a Draw up the sign table for the following function : x ↦−→ 2 x − x 2 b Deduce the definition set of the function f . 2 Consider the function g defined by: g : x ↦−→ x 3 − 3 x − 2 a Expand expression : ( x +1) 2 · ( x − 2) b Determine the definition set of the function g . E.5030 Consider the function f whose image of a number x is defined by the relation: f ( x ) = 3 − 2 · x 2 · x + 3 1 Determine the definition set of the function of the func-tion f . 2 Consider the function g defined on R \ − 3 2 by the rela-tion : g ( x ) = 3 − 2 · x 2 · x + 3 a Determine the reals a and b realizing the following iden-tity: g ( x ) = a + b 2 · x + 3 b Establish the direction of variation of the function g on the interval − 3 2 ; + ∞ . 3 Draw up the table of variations of the function f on its set of definition. E.1484 1 Consider the following two functions : f : x ↦−→ x 2 ; g : x ↦−→ x 2 a Compare the definition sets of the two functions f and g . b What can we say about these two functions on the interval 0 ; + ∞ ? 2 Consider the following functions : h : x ↦−→ x ; j : x ↦−→ x 3 x ; k : x ↦−→ x x a Determine the definition set of each of these functions. b Establish that these functions are all equal on an in-terval. Which interval? E.8165 1 Consider the function f defined on R by: f ( x ) = 6 · x 2 − 11 · x − 10 a Determine the zeros of the function f . b Draw up the table of variations of the function f . (Its zeros should be indicated) 2 Consider the function g defined by: g ( x ) = 6 · x 2 − 11 · x − 10 a Determine the definition set of the function g . b Draw up the table of variations of the function g . E.3079 Consider the function f defined by: f ( x ) = − 2 x 2 − x − 6 1 a Factor the expression : x − 1 2 2 − 25 4 b Justify the following equality: f ( x )= − 2 ( x − 3)( x +2) 2 a Establish the sign table for the expression : ( x − 3)( x + 2) . b Determine the definition set of the function f . 3 Deduce, over the interval 3 ; + ∞ , the monotonicity of the function f . 8. Composition: inverses E.6571 Consider the function f defined by the expression : f ( x ) = 2 · x 2 − 6 · x + 4 We construct a function g defined by the expression : g ( x ) = 1 2 · x 2 − 6 · x + 4 1 Conjecture about the function g : After plotting the representative curve of the function g using the calculator: a Conjecture the set of definition of the function g b Conjecture the direction of variation of the function g on the interval −∞ ; 1 . 2 Study of function g : a Determine the zeros of the function f , then complete the table of variations below : b Justify that the function g is not defined for x =1 and x =2 . https://chingmath.fr −∞.........∞−∞0...0−∞xVariationdef chapExoCorrec/4982 sacados/4982 chapExoCorrec/5030 sacados/5030 chapExoCorrec/1484 sacados/1484 chapExoCorrec/8165 sacados/8165 chapExoCorrec/3079 sacados/3079 chapExoCorrec/6571 sacados/6571 −∞.........∞∞0...0∞xVariationdef
-4-3-2-1234I-3-2-12JOCgCf -∞15∞383-2xVariationdef −∞-238∞∞0−45∞xVariationdeg c Justify that the function g is increasing on −∞ ; 1 . E.7377 Consider the function f defined by the expression : f ( x ) = − x 2 − 2 · x + 3 1 Draw up the table of variations of the function f . Justify the construction of this table and insert the zeros of the function f . 2 a Consider the function g defined by the expression : g ( x ) = − x 2 − 2 · x + 3 Without justification, draw up the table of variations of the function g . b We construct a function h defined by the expression : h ( x ) = 1 − x 2 − 2 · x + 3 Without justification, draw up the table of variations of the function h . E.7303 Consider the two functions f and g de-fined on R \ 2 by the relations : f ( x ) = 1 x − 2 ; g ( x ) = 1 x − 2 2 Without justification, give the direction of variations of the functions f and g on the interval −∞ ; 2 . 9. A little further on - composed of functions E.4977 Consider the two functions f and g de-fined on − 4 ; 4 by the relations : f ( x ) = 2 x + 1 ; g ( x ) = x 2 − 3 We are given the curves C f and C g representing the functions f and g in the coordinate system O ; I ; J ci-dessous : 1 By reading the graph, complete the following tables of values : x − 3 2 − 1 − 1 2 0 1 2 f ( x ) x − 2 − 1 0 1 2 g ( x ) 2 Consider the following calculation program : Prendre a number x ; Determine the image of x by the function f ; we note this number x ; Determine the image of x by the function g ; we note this number g f ( x ) . On peut noter ce programme de calcul par la chaine : x f , −−−−−→ f ( x ) g , −−−−−→ g f ( x ) a Determine the values of the following expressions : g f ( − 1) ; g f − 1 2 b Complete the following table of values : x − 3 2 − 1 − 1 2 0 1 2 g f ( x ) We’ve just created a new function that to a number x asso-ciates the image g f ( x ) . This function is called the com-pound function of f par g and is denoted g ◦ f . 3 Plot the curve C g ◦ f representing the function g ◦ f on the graph above. 4 Give the expression, in terms of x , of the function g ◦ f . E.2719 Consider the two functions f and g de-fined on R whose tables of variation are given below : 1 Determine the value of the following expressions : a g ◦ f (1) b g ◦ f (5) c f ◦ g (8) 2 a Justify that the function g ◦ f is increasing on the interval ] −∞ ; 1] . b Determine the direction of variation of the function g ◦ f on the interval 1 ; 5 3 Draw up the table of variations on R of the composite function g ◦ f . https://chingmath.fr chapExoCorrec/7377 sacados/7377 chapExoCorrec/7303 sacados/7303 chapExoCorrec/4977 sacados/4977 -4-3-2-1234I-3-2-12JOCgCf chapExoCorrec/2719 sacados/2719 -∞15∞383-2xVariationdef −∞-238∞∞0−45∞xVariationdeg
-4-3-2-1234I-4-3-2-123JOCf -3-2-123I-3-2-123JOCf -3-2-123I-3-2-123JOCg -∞02∞∞−230Variationdefx E.2234 Consider the function f defined on the interval − 4 ; 4 whose representative curve C f is given below in the ( O ; I ; J ) orthonormal coordinate system : 1 Calculate the following images : a f ◦ f (1) b f ◦ f ( − 2) c f ◦ f (3) 2 We define the function f n as the function composed n times of the function f by itself. Determine the value of the following images : a f 3 (1) b f 3 ( − 3) c f 4 ( − 1) E.2698 Consider two functions f and g defined on the interval [ − 3 ; 3] whose representative curves, respec-tively C f and C g , are given in a reference frame O ; I ; J orthonormal : 1 Determine the value of the following expressions : a f ◦ g ( − 2) b f ◦ g (1.5) c f ◦ g (2) 2 Determine the value of the following expressions : a g ◦ f ( − 3) b g ◦ f (0) c g ◦ f (1) E.4979 Consider the following two functions : f : x ↦−→ x 2 − 2 ; g : x ↦−→ x 1 a Give the definition sets of the functions f and g . b Can we talk about the image of 1 by the function g ◦ f ? Justify your answer. 2 a Draw up the sign table for the function f . b Give the definition set of the function g ◦ f . E.2327 Consider the function f defined on R whose table of variations is given below : Draw up the table of variations of the functions g , h , j defined below : 1 g ( x ) = f ( x +2) 2 h ( x ) = f (3 x ) 3 j ( x ) = f (2 x +1) E.4978 For each of the pairs of functions below, determine the expression of the function g ◦ f : 1 f : x ↦−→ 3 x − 5 ; g : x ↦−→ x 2 2 f : x ↦−→ x 2 − 1 ; g : x ↦−→ − 2 x + 4 3 f : x ↦−→ x 2 + 1 ; g : x ↦−→ x 2 + 1 10. Unclassified financial years E.409 Three functions are considered : f : x , −−−−−→ 3 x − 1 ; g : x , −−−−−→ x 2 h : x , −−−−−→ 2 x − 1 Each of these functions are reference functions. 1 Consider the composite function f by g and then by h . That is, the image of 2 is calculated by this compound function as follows : 2 f , −−−−−→ 5 g , −−−−−→ 25 h , −−−−−→ 49 a Calculate the image by this function of the numbers − 1 , 0 and 1 . b Calculate the antecedents of 127 . c Give an algebraic writing of this compound function. 2 We’re going to do the reverse job. Here are some new functions, break them down into reference functions. a k ( x ) = 3 x 2 + 1 b ‘ ( x ) = 1 3 x + 1 c m ( x ) = 1 − x 2 d n ( x ) = 3 x + 5 2 https://chingmath.fr chapExoCorrec/2234 sacados/2234 -4-3-2-1234I-4-3-2-123JOCf chapExoCorrec/2698 sacados/2698 -3-2-123I-3-2-123JOCf -3-2-123I-3-2-123JOCg chapExoCorrec/4979 sacados/4979 chapExoCorrec/2327 sacados/2327 -∞02∞∞−230Variationdefx chapExoCorrec/4978 sacados/4978 chapExoCorrec/409 sacados/409
-∞-13∞∞52∞xVariationdef -∞3−√2910310√2910∞-∞029200∞VariationdeQx E.2825 1 Let f be a function defined on R . Consider the function : g : x ↦−→ − 2 · f ( x ) + 5 a If f is increasing, justify that the function g is decreas-ing. b If f is decreasing, justify that the function g is increas-ing. 2 Consider the function h defined by the relation: h ( x ) = − 2 x 2 + 5 Deduce, from the previous question, that the function h is increasing on R − . 3 Justify each of the following assertions : a j : x ↦−→ 5 − 2 x is increasing on R ∗ + . b k : x ↦−→ 2 · ( x + 2) 2 − 5 is increasing on − 2 ; + ∞ . E.2849 Consider a function f defined on R whose table of variations is as follows : Consider the function g whose image of a number x is defined by the relation: g ( x ) = 2 f ( x ) − 1 + 3 1 Determine the image of the numbers − 1 and 3 by the function g . 2 Determine the direction of variation of the function g on the interval ] −∞ ; 3] . E.2720 Let g be the function defined by the relation: g ( x ) = 1 − 5 x 2 + 3 x + 1 1 Determine the definition set of this function. 2 Justify that the polynomial Q = − 5 x 2 +3 x +1 admits the following table of variations : 3 Draw up the table of variations of the function g . (Only directions of variation will be shown) E.5056 Consider the function f defined on R by: f ( x ) = 1 x + 1) 2 + 2 Determine the direction of variation of the function f on the interval − 1 ; + ∞ . E.2717 Consider the function f defined on R ∗ defined by the relation: f ( x ) = 2 · 1 x 2 − 3 · 1 x + 5 1 Draw up the table of signs and the table of variations of the polynomial of the second degree : P ( x ) = 2 x 2 − 3 x + 5 2 Noting that the function f is the compound of the in-verse function with this polynomial of the second degree, establish that the function is decreasing 0 ; 4 3 3 Draw up the table of variations of the function f (do not attempt to complete the table with the values of the images.) E.2718 Consider the function f whose image of x is defined by the relation: f ( x ) = − x 2 + 2 x + 1 1 Determine the definition set of the function f . 2 Draw up the table of variations of the function f . https://chingmath.fr chapExoCorrec/2825 sacados/2825 chapExoCorrec/2849 sacados/2849 -∞-13∞∞52∞xVariationdef chapExoCorrec/2720 sacados/2720 -∞3−√2910310√2910∞-∞029200∞VariationdeQx chapExoCorrec/5056 sacados/5056 chapExoCorrec/2717 sacados/2717 chapExoCorrec/2718 sacados/2718