Grade 10 / Study of functions 36 exercises (100% corrected)

a
-5-4-3-2-123I-2-123456JOCf ChingQuizz : 1 exercise available for Quizz assessment : 1. Framing by an affine function E.9403 Let x be a real number with frame 2 x 4 . For each question, determine a frame for the proposed literal expression : a 2 x + 1 b − x + 3 c 2 x d 3 x − 5 E.4975 Let x be a real number whose frame is − 1 <x 3 . For each question, determine a frame for the proposed literal expression : a 3 x + 1 b − 2 x − 5 E.9407 Let x be a real number whose frame is − 1 <x 3 . For each question, determine a frame for the proposed literal expression : a 3 x + 1 b − 2 x − 5 c x + 2 d 1 − 2 · x 2. Framing by a function of the second degree E.8283 Let x be a real number with frame 2 x 4 . For each question, determine a frame for the proposed literal expression : a x 2 b 2 − x 2 c x − 4 2 d ( x + 1) 2 + 1 E.8282 Let x be a real number with frame 1 <x 4 . For each question, determine a frame for the proposed literal expression : a x − 5 2 − 2 b ( x + 1) 2 + 1 E.338 Let x be a real number such that 0 <x 2 . Give the frames of the following numbers : a ( x − 2) 2 b − 2 · x 2 − 1 c ( x + 1) 2 + 1 E.9371 Let x be a real number verifying the frame 2 x< 5 . For each question, determine a frame for the proposed literal expression : a 1 − x 2 b x − 4 2 E.412 Given the expression 2 x 4 , deter-mine the expressions for the following algebraic expressions : a 3 x − 4 2 + 2 b x 2 + 1 d 2 x − 6 2 3. Square function: study of second degree functions E.2846 Consider the function f defined on − 5 ; 3 whose representation is given below : https://chingmath.fr chapExoCorrec/9403 sacados/9403 chapExoCorrec/4975 sacados/4975 chapExoCorrec/9407 sacados/9407 chapExoCorrec/8283 sacados/8283 chapExoCorrec/8282 sacados/8282 chapExoCorrec/338 sacados/338 chapExoCorrec/9371 sacados/9371 chapExoCorrec/412 sacados/412 chapExoCorrec/2846 sacados/2846 -5-4-3-2-123I-2-123456JOCf
The following questions are to be answered graphically and without justification. 1 Draw up the table of variations of the function f . 2 Draw up the table of signs of the function f . 3 Solve the inequation : f ( x ) > 5 2 4 Determine the images of the following intervals : a 1 ; 3 b − 4 ; − 2 c − 15 4 ; 1 4. Framing by a rational function E.7375 Let x be a real number verifying the frame 2 x< 5 . For each question, determine a frame for the proposed literal expression : a 3 2 · x − 3 E.9369 Let x be a real number with frame 1 <x 4 . For each question, determine a frame for the proposed literal expression : a 1 x b 1 x + 2 c 1 − 1 x + 2 E.9370 Let x be a real number whose frame is − 1 <x 3 . For each question, determine a frame for the proposed literal expression : a 1 x + 2 b 1 x 2 + 1 c 2 ( x − 1) 2 + 1 E.350 Let x be a real number whose frame is − 1 <x 3 . For each question, determine a frame for the proposed literal expression : a 1 x + 2 b 1 x 2 + 1 c 2 ( x − 1) 2 + 1 E.9408 From the frame 2 x 4 , deduce the frames of the following algebraic expressions : a 1 x + 2 b 1 x 2 c x 2 2 x + 1 5. Inverse function: study of homographic function E.7862 Consider the function f defined by: f : x ↦−→ 3 x + 1 x − 3 1 Show that, for any x ∈ R \ 3 , we have : 3 x + 1 x − 3 = 3 + 10 x − 3 2 Determine the direction of variation of the function f on the interval −∞ ; 3 . E.4984 Consider the function g defined on R \ 2 by: g ( x ) = x + 1 2 − x 1 Determine the reals a and b realizing the identity: g ( x ) = a + b 2 − x 2 Draw up the table of variations of the function g E.9406 Let f be the function defined on R \ 3 2 by the relation: f ( x ) = 8 x − 11 2 x − 3 1 Determine the reals a and b realizing the identity: f ( x ) = a + b 2 x − 3 2 Draw up the table of variations of the function f . E.8163 Consider the function f defined by: f ( x )= 8 − 5 · x x − 2 1 Determine the values of the real numbers a and b realiz-ing the identity below : f ( x ) = a + b x − 2 2 Establish that the function f is increasing on the interval −∞ ; 2 . E.6573 Consider the function f defined on R by the expression : f ( x ) = 2 · x 2 + 4 · x + 1 x 2 + 2 · x + 2 1 Determine the two real a and b verifying the identity: f ( x ) = a + b x + 1 2 + 1 2 a Determine the direction of variation of the function f on the interval − 1 ; + ∞ . b Show that the function f is decreasing on the interval −∞ ; − 1 . https://chingmath.fr chapExoCorrec/7375 sacados/7375 chapExoCorrec/9369 sacados/9369 chapExoCorrec/9370 sacados/9370 chapExoCorrec/350 sacados/350 chapExoCorrec/9408 sacados/9408 chapExoCorrec/7862 sacados/7862 chapExoCorrec/4984 sacados/4984 chapExoCorrec/9406 sacados/9406 chapExoCorrec/8163 sacados/8163 chapExoCorrec/6573 sacados/6573
AB-1IJOCf E.4877 Consider the function f defined on R by the relation: f ( x ) = − 4 · x 2 − 2 x 2 + 1 1 Demonstrate the following equality: f ( x )= 2 x 2 +1 − 4 2 a Show that the function f is strictly increasing on the interval −∞ ; 0 . b Establish the direction of variation on the interval 0 ; + ∞ of the function f . c Draw up the table of variations of the function f (the values of the local extremums will be indicated) . 3 From the previous question, deduce the maximum value of the function f . 4 a Algebraically determine, the antecedents of the num-ber − 3 by the function f . b Solve the following equation : f ( x )= − x − 2 6. Square root: use E.4973 Let f be the function whose image of a number x is defined by the relation: f ( x ) = − 2 x + 1 + 3 1 Justify that the definition set of the function f is : D f = − 1 ; + ∞ 2 Establish that the function f is decreasing on its defining set. E.4974 Consider the function f defined by the relation: f ( x ) = − 2 x + 3 1 Justify that the defining set of f is R + . 2 Establish that f is strictly increasing on R + . E.8166 Consider the function f defined by: f ( x ) = − x 2 − 2 · x + 7 9 1 a Determine the definition set of the function f . b Draw up the table of variations of the function f . 2 In the plane provided with a reference frame O ; I ; ;J , consider the curve C f representative of the function f and the circle C of center A ( − 1 ; 0) and radius 4 3 . a Show that any point M of the curve C f belongs to the circle C . b Consider the semicircle E formed by the set of points on the circle C whose ordinate is positive or zero. This subset can be written as : E = M ( x ; y ) ∈ C ⏐ ⏐ y > 0 Conversely, justify that each point of E belongs to the curve C f representative of the function f . E.7380 Consider the function f defined on the interval − 1 ; 1 by the relation: f ( x ) = 1 − x 2 whose representative curve C f is given below in a reference frame O ; I ; J below : Let x be a real number belonging to the interval 0 ; 1 . Note A and B respectively, the points on the curve C f with abscis-sas x and − x respectively. Determine the abscissa of point A so that the area of triangle OAB is maximum. Hint: we can establish the identity: x 2 − x 4 = 1 4 − x 2 − 1 2 2 7. Cube function: use E.8281 Consider the function f defined on R by: f ( x ) = x 3 + 6 · x 2 + 12 · x + 12 1 Establish that for any real number x , we have : f ( x ) = x + 2 3 + 4 2 Establish that the function f is increasing on R . 8. Algebraic study of variations https://chingmath.fr chapExoCorrec/4877 sacados/4877 chapExoCorrec/4973 sacados/4973 chapExoCorrec/4974 sacados/4974 chapExoCorrec/8166 sacados/8166 chapExoCorrec/7380 sacados/7380 AB-1IJOCf chapExoCorrec/8281 sacados/8281
E.7335 Consider the function f defined on R by the relation: f ( x ) = x 3 + x + 2 1 Let a and b be any two real numbers. Determine the identity: f ( a ) − f ( b ) = a − b · b 2 + a · b + a 2 + 1 2 Establish that the function f is strictly increasing on −∞ ; 0 E.8312 Consider the function f defined on R \{ 1 by the relation: f ( x ) = 1 x − 1 1 Show that the function f is decreasing on 1 ; + ∞ 2 a Let a and b be two distinct real numbers of 1 . Show that : f ( a ) − f ( b ) = b − a a − 1 b − 1 b Deduce that the function f is decreasing on −∞ ; 1 E.8164 Consider the function f defined by the expression : f ( x ) = 1 x 2 + 1 1 Determine the definition set of the function f . 2 a For all real numbers a and b , establish the following identity: f ( b ) − f ( a ) = a − b a + b a 2 + 1 b 2 + 1 b Deduce that the function f is increasing on the interval R − . E.7376 Consider the function f defined by the relation: f ( x ) = 2 · x 2 + x + 1 1 Justify that the function f is defined on R . 2 a Let a and b be two real numbers, establish the iden-tity: f ( b ) − f ( a ) = b − a · 2 · b + 2 · a + 1 2 · b 2 + b + 1 + 2 · a 2 + a + 1 b Deduce that the function f is increasing on the interval − 1 4 ; + ∞ 9. A little further - definition set E.2141 Consider the following two func-tions : f : x ↦−→ 2 x − 1 · 4 x + 3 ; g : x ↦−→ 2 x − 1 4 x + 3 1 a Using a graphing calculator, draw the representative curve of the function f , then that of the function g . b Graphically, give the interval on which these two func-tions coincide. 2 Determine the definition set of the function f . 3 a Determine the sign table of the algebraic expression (2 x − 1)(4 x +3) . b Deduce the domain of definition of the function g . E.536 1 Expand the expression : ( x +1)(3 − 2 x )( x − 1) . 2 Consider the two functions f and g whose images of the number x are defined as follows : f ( x ) = − 2 x 3 + 3 x 2 + 2 x − 3 x 2 − 1 ; g ( x ) = 3 − 2 x a Determine the definition set of each of these functions. b Establish that the two functions f and g are equal on a set to be specified. E.2194 1 Consider the two functions f and g whose image of x is defined by the relations : f ( x ) = − x 2 − 3 x + 4 (3 x − 2)( − x + 1) ; g ( x ) = x + 4 3 x − 2 a Simplify the expression : g ( x ) − f ( x ) . b Compare the functions f and g . 2 Consider the two functions j and ‘ defined by: j ( x ) = 1 3 x + 1 + 4 − x ; ‘ ( x ) = 3 x + 1 − 4 − x 4 x − 3 a Determine the definition set of these two functions. b Expand the following expression : A = 3 x + 1 + 4 − x 3 x + 1 − 4 − x c Show that the two functions j and ‘ are equal on the set − 1 3 ; 4 \ 3 4 E.2694 1 Consider the two functions : f : x ↦−→ 3 x − 2 ; g : x ↦−→ 14 x + 7 2 x + 1 a Establish the following equality: 6 x 2 + 13 x + 5 = (2 x + 1)(3 x + 5) b Consider the function f + g . Establish equality: f + g ( x ) = 3 x + 5 c Give the set on which the function f + g is defined. 2 Consider the following two functions : f : x ↦−→ 2 x + 1 x 2 + x ; g : x ↦−→ ( x + 1)( − 3 x + x 2 ) a Consider the function f · g . Establish the following equality: f × g ( x ) = 2 x 2 − 5 x − 3 b Give the set on which the function f · g is defined. 3 Consider the two functions f and g defined by: f : x ↦−→ x 2 − x ; g : x ↦−→ x + x a Establish the following equality: f g ( x ) = x − x b Give the set on which the function f g is defined. https://chingmath.fr chapExoCorrec/7335 sacados/7335 chapExoCorrec/8312 sacados/8312 chapExoCorrec/8164 sacados/8164 chapExoCorrec/7376 sacados/7376 chapExoCorrec/2141 sacados/2141 chapExoCorrec/536 sacados/536 chapExoCorrec/2194 sacados/2194 chapExoCorrec/2694 sacados/2694
10. Framing and direction of variation E.1904 Let x be a real number: 1 Suppose 1 <x 3 . Give a frame for each of the following numbers : a x 2 b 1 − 1 x c 3 − 2 x 2 Suppose − 4 <x< 1 and consider y a real number verify-ing the frame 2 <y< 4 . a Give a frame for the following expressions : x + y ; ( x + 4) · y b Justify, with a counterexample, that the framing below is false : − 8 <x · y < 4 E.2832 Let x be a number verifying the frame : 3 <x 5 Determine a frame for each of these expressions : a 2 · 1 2 · x − 3 2 − 2 b 1 (2 − x ) 2 11. Unclassified exercises E.1941 1 Let a and b be two numbers belonging to the interval − 2 ; 0 such that a<b . Make the following comparison : 3 a 2 +1 < 3 b 2 +1 2 Deduce the direction of variation, on the interval [ − 2 ; 0] , of the function f defined by: f : x ↦−→ 3 x 2 +1 https://chingmath.fr chapExoCorrec/1904 sacados/1904 chapExoCorrec/2832 sacados/2832 chapExoCorrec/1941 sacados/1941