Outside the high school program / Odd/even function 27 exercises (including 21 corrected)

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ijIMN -5-4-3-2-123456I-3-2-123456JOC 1. Introduction E.2165 Theoretical study: 1 Consider a function f whose representative curve C f ad-mits the point I of coordinates ( a ; b ) as center of symme-try. Let’s take two points M ( x ; y ) and N ( x ; y ) of the curve C f symmetrical with respect to the point I . The graph below illustrates this situation : It is assumed that the points M and N are such that : x<a<x (even if it means reversing the role of M and N ) a Using the fact that the point I is the midpoint of the segment [ MN ] , express the coordinates of the point I in terms of the coordinates of the points M and N . b We pose h = x − a , deduce the following relationship : f ( a + h ) + f ( a − h ) 2 = b 2 Let’s take the opposite approach : we’ll show that un-der certain conditions, the curve of a function admits a center of symmetry. Assume the existence of a function g and a point I ( a ; b ) such that : For any h ∈ R such that ( a + h ) ∈D g then ( a − h ) ∈D g g ( a + h ) + g ( a − h ) 2 = b In the rest of the exercise, let’s assume that h is a real number such that ( a + h ) ∈D g . Let us denote N the point of C g of abscissa ( a + h ) : a What property of the function g allows us to state that the curve C g admits a point of abscissa ( a − h ) ? We will note M the point on the curve of C g having ab-scissa ( a − h ) . b Let’s note M this point ; establish that the coordinates of point N are: M ( a − h ; 2 · b − g ( a + h )) c Show that the point I is the midpoint of the segment [ MN ] . d Justify that the curve C g admits the point I as center of symmetry. Study of a function : Consider the function ‘ defined on R \{ 2 } by the relation: ‘ ( x ) = 3 x + 2 2 x − 1 Here is the representative curve of this function : 3 What conjecture can be made about a geometric prop-erty of this curve? 4 a Show that : ‘ 1 2 + h = 7 4 h + 3 2 . b Show that : ‘ 1 2 − h = − 7 4 h + 3 2 c Deduct the value of : ‘ 1 2 − h + ‘ 1 2 + h 2 d What can we deduce about the curve C ? 2. Parity study E.3315 Study the parity of the functions below : a f ( x ) = x − 1 · x + 1 b g ( x ) = 3 x 3 − 2 x c h ( x ) = x 2 + 1 x d j ( x ) = cos 3 · x 3 E.3609 1 Justify that the following function is even : f : x ↦−→ e x + e − x 2 Justify that the following function is odd : g : x ↦−→ e x − e − x https://chingmath.fr chapExoCorrec/2165 sacados/2165 ijIMN -5-4-3-2-123456I-3-2-123456JOC chapExoCorrec/3315 sacados/3315 chapExoCorrec/3609 sacados/3609
3. Axis of symmetry of curves E.3316 1 Consider the function f defined on R whose image of x is defined by: f ( x ) = 2 x 2 + 4 x − 1 We note C f the representative curve of the function f . a Let h be a real number. Determine the simplified ex-pressions as a function of h of : f ( − 1 − h ) ; f ( − 1 + h ) b Deduce a geometric property of the curve C f . 2 Consider the function g defined on R \{ 1 } defined by: g ( x ) = x 2 − x + 1 x − 1 Justify that the curve C g , representative of the function g , admits the point of coordinate (1 ; 1) for center of sym-metry. 4. Center of symmetry of curves E.2302 Consider the function f defined on R \{ 2 } by the relation: f ( x ) = x 2 − 4 x + 7 2 x − 4 Show that the point I (2 ; 0) is the center of symmetry of the curve C f E.2807 Consider the function f defined on R − {− 3 } whose image of x is given by the relation: f ( x ) = x 2 + 10 x + 20 2 x + 6 Show that the curve C f representative of the function f ad-mits for center of symmetry the point K ( − 3 ; 2) E.2520 Consider the function f defined by the relation: f : x ↦−→ 6 x + 4 3 x + 1 1 Give the definition set of this function. 2 a Establish the relationship : f ( x ) = 2 3 x + 1 + 2 b Deduce the writing of the function g verifying that the following relationship for any x ∈D f : f ( x ) = g (3 x + 1) 3 Establish that the representative curve C f admits as cen- ter of symmetry the point of coordinate − 1 3 ; 2 4 Consider the straight line ( d ) of equation y = x +2 . Determine the set of abscissas of the points , as a reunion of intervals, on which ( d ) lies above C f . E.3985 Let f be the function defined on R by: f ( x ) = x + 2 − 4 · e x e x + 3 We denote by C its representative curve in the plane referred to an orthonormal reference O ; −→ i ; −→ j . Let I be the point of C with abscissa ln 3 . Show that the point I is the center of symmetry of the curve C . E.4300 Consider the function f defined on R by: f ( x ) = 4 · e x e x + 7 We denote by C the representative curve of the function f in an orthonormal reference frame O ; −→ i ; −→ j . 1 Verify that for any real x , we have : f ( x )= 4 1+7 · e − x 2 Show that the point I 1 of coordinates (ln 7 ; 2) is a center of symmetry of the curve C . 3 Determine an equation of the tangent ( T ) to the curve C at the point I . 5. Axis and center of symmetries of curves E.2171 1 Show that the function f defined on R by: f ( x ) = 6 1 2 x 2 + x + 2 admits the line of equation x = − 1 as its axis of symme-try. 2 Show that the function g defined on R \{− 1 } by: g ( x ) = x 2 + 4 x + 2 x + 1 admits the point I ( − 1 ; 2) as center of symmetry. https://chingmath.fr chapExoCorrec/3316 sacados/3316 chapExoCorrec/2302 sacados/2302 chapExoCorrec/2807 sacados/2807 chapExoCorrec/2520 sacados/2520 sacados/3985 sacados/4300 chapExoCorrec/2171 sacados/2171
-3-2-123I-123JO E.2198 1 Establish that the representative curve of the function f defined on R \{ 2 } by: f ( x ) = 2 x + 1 2 − x admits point I (2 ; − 2) as center of symmetry. 2 Let g be a function defined on R whose image of x is given by the relation: g ( x ) = 2 x 2 + 4 x − 4 − x 2 − 2 x − 3 Show that the curve C g admits the straight line with equation x = − 1 as its axis of symmetry. E.2216 1 Consider the function f defined on [ − 5 ; 1] whose image of x is defined by the relation: f ( x ) = − x 2 − 4 x + 5 Show that the curve C f admits the straight line of equa-tion x = − 2 as axis of symmetry. 2 Consider the function g defined on R \{− 1 } by the rela-tion : g : x ↦→ x 2 + 3 x + 3 x + 1 Show that the curve C g admits the point ( − 1 ; 1) as center of symmetry. 6. Symmetry and oblique asymptotes E.3366 Consider the function f whose image of x is defined by the relation: f ( x ) = 3 x 3 + 4 x 2 − 5 x + 4 4( x 2 + 1) In the plane equipped with a coordinate system O ; I ; J , we denote C f the curve representing the function f . 1 a Determine the values of the real numbers a , b , c that satisfy the relation: f ( x ) = a · x + b + c · x x 2 + 1 b Show that the curve C f has an oblique asymptote (Δ) whose equation will be specified. c Study the relative position of the curve C f and the line (Δ) . 2 Establish that the curve C f has the point with coordi-nates (0 ; 1) as its center of symmetry. 3 a Établir que la dérivée f de la fonction f admet pour dérivée : f ( x ) = x 2 + 5 3 x 2 − 1 4 x 2 + 1 2 b Draw the table of variations of the function f . Remark : we’ll admit the following two results : f 3 3 = 1 − 3 4 ≈ 0.57 f − 3 3 = 1 + 3 4 ≈ 1.43 4 Curve C f . 7. Symmetries and derivatives E.2392 Consider the function f defined by the relation: f ( x ) = 3 x + 6 2 x 2 + 8 x + 7 1 Determine the definition set of the function f . 2 Show that the representative curve of the function f ad-mits as center of symmetry the point A of coordinate A ( − 2 ; 0) . 3 Show that the expression of the number derived from f in x is expressed by the relation: f ( x ) = − 3 · 2 x 2 + 8 x + 9 (2 x 2 + 8 x + 7) 2 4 Establish that the representative curve of the function f admits as axis of symmetry the line of equation x = − 2 . https://chingmath.fr chapExoCorrec/2198 sacados/2198 chapExoCorrec/2216 sacados/2216 chapExoCorrec/3366 sacados/3366 -3-2-123I-123JO chapExoCorrec/2392 sacados/2392
-4-224I-4-224JO E.3505 Let f be a function f defined on R verifying the following limit: lim h ↦→ 0 f (2+ h ) − f (2) h = − 1 2 1 What can be said about the derivability of the function f in 2 . 2 Suppose the function f is even : a Determine the number derived from the function f in − 2 . b Freehand, represent a curve C f and its two tangents at − 2 and 2 verifying such a situation. 3 Suppose the function f is odd : a Determine the derivative number of the function f in − 2 . b Freehand, represent a curve C f and its two tangents at − 2 and 2 verifying such a situation. E.2393 Consider the function f whose image of x is defined by the relation: f ( x ) = − 6 x − 1 3 x + 1 1 Determine the definition set of the function f . 2 a Show the following equality: f ( x )= 1 3 x +1 − 2 b Deduce that the representative curve C f admits as cen-ter of symmetry the point I of coordinate − 1 3 ; − 2 . 3 Give the expression of the derivative function of the func-tion f . 4 Show that the representative curve C f of the derivative function f admits as axis of symmetry the straight line of equation x = − 1 3 . 8. Symmetries and integrals E.4221 Consider the function f defined on − 1 ; 1 by: f ( x ) = 3 2 · ⏐ ⏐ x ⏐ ⏐ − 1 2 1 Study the parity of the function f . 2 Show that this function is the density of a probability law on − 1 ; 1 . 9. Curve symmetries E.411 1 a In the marker below, place the points A ( − 3 ; 4) , B (4 ; 2) and C ( − 1 ; − 2) . b Place the symmetries of the points A , B , C with re-spect to the axis ( yy ) and with respect to the origin O of the reference frame. c Complete the following table : Coordonnées of the point of the image by ( yy ) of the image by O A ( − 3 ; 4) B (4 ; 2) C ( − 1 ; − 2) d Complete the following sentences : The points ( x ; y ) and ( − x ; y ) are symmetrical about the axis ( yy ) if, and only if, : : : : : : The points ( x ; y ) and ( − x ; y ) are symmetrical about the origin O if, and only if, : : : : : : 2 Consider the following functions : f : x ↦−→ x 2 ; g : x ↦−→ x 3 − x ; h : x ↦−→ | 2 x − 1 | a Complete the following table : x − 3 − 2 − 1 0 1 2 3 ( x ; f ( x )) ( x ; g ( x )) ( x ; h ( x )) https://chingmath.fr chapExoCorrec/3505 sacados/3505 chapExoCorrec/2393 sacados/2393 sacados/4221 sacados/411 -4-224I-4-224JO
0IJ b Make a conjecture as to whether the curves C f , C g and C h admit the straight line ( yy ) as axis of symmetry or the origin of the reference frame as center of symmetry. c Draw the representative curves of these functions on your calculator. 3 Algebraic study of the functions f and g : a Express f ( − x ) as a function of x . Simplify the writing of f ( − x ) . What do we notice? b Express g ( − x ) as a function of x . Simplify the writing of g ( − x ) . What do we notice? E.413 For each of the following functions, give their definition sets and then study their parities : a f : x , −−−−−→ 1 − x 2 b g : x , −−−−−→ | x | x ( x 2 − 1) c h : x , −−−−−→ 3 x 2 − x + 1 d j : x , −−−−−→ 3 x ×| x | E.415 Consider the function f defined on [0 ; 8] , for which we know only the following table of variations : x 0 1 2 3 4 5 6 7 8 f ( x ) 0 1 2 ; 5 1 − 2 − 0.5 0 1 2 We further assume that this function is strictly monotonic on each of the following intervals : [0 ; 2] , [2 ; 4] , and [4 ; 8] . 1 Construct the variation table for the function f . 2 Give a bound for f ( x ) in each of the following cases : a 0 x 4 b 4 <x < 8 c 2 x < 7 3 Plot, in black, a possible graph of the function f on the coordinate plane below. 4 In this question, consider the function f 1 defined on [ − 8 ; 8] as the even extension of the function f : a Complete the table below correctly: x − 8 − 7 − 6 − 5 − 4 − 3 − 2 − 1 f 1 ( x ) b Find the set of antecedents of 1. c Plot the graph of f 1 in red on the coordinate plane below. 5 In this question, we consider the function f 2 defined on [ − 8 ; 8] as the odd extension of the function f : a Complete the table below correctly: x − 8 − 7 − 6 − 5 − 4 − 3 − 2 − 1 f 2 ( x ) b Consider the following points in the plane : A ( − 4 ; 2) ; B (3 ; 1) ; C (4 ; − 2) ; D ( − 3 ; − 1) Prove that the quadrilateral ABCD is a parallelogram. c In the coordinate system below, plot the graph of f 2 in green. E.422 1 Let f be a function defined on an interval I centered at O . The following two functions are defined : g : x ↦−→ f ( x ) + f ( − x ) 2 ; h : x ↦−→ f ( x ) − f ( − x ) 2 Study the parity of the function g and h . 2 Let h be an odd function defined on D h such that 0 ∈ D h . Show that : h (0)=0 E.423 Give the domain and parity of the follow-ing functions : a f : x ↦−→ x ( x + 2) 2 b g : x ↦−→ x 2 − 1 c h : x ↦−→ 1 ( x − 4)( x + 4) d j : x ↦−→ 1 2 x · | x | E.1754 We place ourselves in an orthogonal frame of reference ( O ; I ; J ) 1 a Let M be a point in the plane with coordinates ( x ; y ) . Give the coordinates of the image of point M by or-thogonal symmetry of axis ( OJ ) . b Let f be a function defined on R , verifying for any real number x : f ( x ) = f ( − x ) Show that the function f is even. 2 a Let M be a point in the plane with coordinates ( x ; y ) . Give the coordinates of the image of point M by cen-tral symmetry of center O . b Let f be a function defined on R , verifying for any real number x : f ( x ) = − f ( − x ) Show that the function f is odd. https://chingmath.fr chapExoCorrec/413 sacados/413 chapExoCorrec/415 sacados/415 0IJ chapExoCorrec/422 sacados/422 chapExoCorrec/423 sacados/423 sacados/1754
-8-6-4-202468-4-224C -8-6-4-202468-4-224 -8-6-4-202468-4-224Ch -8-6-4-202468-4-224Ck -8-6-4-202468-4-224C -2-12345678910I-5-4-3-2-1234JO E.1759 1 For each function, calculate the requested images : Calculate f ( − 6) , f ( − 2) , f (0) , f (2) and f (6) . Calculate g ( − 7) , g ( − 1) , g (0) , g (1) and g (7) . Calculate h ( − 3) , h (0) and g (3) . Calculate k ( − 3) , k ( − 2) , k (0) , k (2) and k (3) . 2 For each of the following functions say whether it is even, odd or neither. 10. Curve symmetry E.426 Consider the function f defined by: f : x ↦−→ 1 4 · x 2 − 2 · x − 1 1 Using the calculator, complete the table below with val-ues rounded to the nearest hundredth : x − 2 − 0.5 1 2 3 3.5 4 f ( x ) x 4.5 5 6 7 8.5 10 f ( x ) 2 Draw the curve C f in the reference frame O ; I ; J be-low : 3 This curve has an axis of symmetry; draw this axis on your representation. https://chingmath.fr sacados/1759 -8-6-4-202468-4-224C -8-6-4-202468-4-224 -8-6-4-202468-4-224Ch -8-6-4-202468-4-224Ck -8-6-4-202468-4-224C chapExoCorrec/426 sacados/426 -2-12345678910I-5-4-3-2-1234JO
E.8187 In an orthonormal coordinate system O ; I ; J , consider the curve C f representing the function f defined by a second-degree polynomial. The only information we have about the function f is the table of values below : x − 5 − 3 − 2 0 2 4 f ( x ) 36 0 − 9 − 9 15 63 1 Give the equation of the axis of symmetry of the curve C f . 2 Give the two antecedents of the number 0 by the function f . 3 Give the expression of the function f . E.404 Consider the function f whose image of a real number x is defined by: f ( x ) = − 2 x 2 + 4 x + 3 1 a Draw up the table of variations of the function f . b Specify the characteristics of the extremum of the func-tion f . 2 Let h be any positive number: a Determine the developed and reduced form of the fol-lowing two expressions : f (1 − h ) ; f (1+ h ) b What can be said about the two points of C f of respec-tive abscissas (1 − h ) and (1+ h ) . https://chingmath.fr chapExoCorrec/8187 sacados/8187 chapExoCorrec/404 sacados/404