Outside the high school program / Absolute values 30 exercises (including 24 corrected)

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ABCDx0IyJ ABCx0IyJ EFGx0IyJ JKLMx0IyJ x0IyJ ABCDOM6cm4cm 1. Piecewise affine functions E.428 In the orthonormal reference frame O ; I ; J below, consider the polygonal line formed using the four plane points A , B , C and D . 1 a Give the three equations of the straight lines carried respectively by the segments [ AB ] , [ BC ] and [ CD ] . This polygonal line defines a function f defined on [ − 5 ; 5] say piecewise affine function (or by intervals) , be-cause on each of its three intervals : I =[ − 5 ; − 1[ ; J =[ − 1 ; 1[ ; K =[1 ; 5] the representative curve of f is an affine function. Algebraically, we define the function f as follows : f : f ( x ) = − x − 5 si x ∈ [ − 5 ; − 1] f ( x ) = 2.5 x − 0.5 si x ∈ [ − 1 ; 1] f ( x ) = − 0.5 x + 2.5 si x ∈ [1 ; 5] b Give the images of the numbers − 4 , − 1 , 2 and 5 by the function f . 2 Here are the three graphical representations of the piece-wise affine functions g , h and j . a Give the definition set of each of these functions. b Give their algebraic expressions. E.420 Plot the representative curve of the follow-ing function in the frame below : f : f ( x ) = 2 x + 8 si x ∈ [ − 5 ; − 2] f ( x ) = − 0.5 x + 3 si x ∈ ] − 2 ; 2] f ( x ) = − 2 x + 6 si x ∈ ]2 ; 4] E.3000 In the plane, consider the rectangle ABCD of length 5 cm and width 3 cm . A point M traverses the contour of this rectangle; this point is marked by the num-ber x representing the distance traveled by this point starting from A and traversing the rectangle in the direct direction. Note A ( x ) the area of the hatched part when M is marked by the number x . 1 Justify that the function A is a piecewise affine function as a function of x defined by the system : x for x ∈ [0 ; 6] 3 2 x − 3 for x ∈ [6 ; 10] x + 2 for x ∈ [10 ; 16] 3 2 · x − 6 for x ∈ [16 ; 20] 2 Determine the value of x for which we have : A ( x ) = 20 cm 2 2. Absolute value - algebraic calculation https://chingmath.fr sacados/428 ABCDx0IyJ ABCx0IyJ EFGx0IyJ JKLMx0IyJ sacados/420 x0IyJ sacados/3000 ABCDOM6cm4cm
-4-3-2-12345I-3-2-123JO E.320 Carry out the calculations below, ex-plaining your approach : a ⏐ ⏐ 2 3 − 3 2 ⏐ ⏐ b ⏐ ⏐ ⏐ 1 ı − 1 3 ⏐ ⏐ ⏐ c ⏐ ⏐ ⏐ 5 8 4 − 8 5 5 ⏐ ⏐ ⏐ 3. Absolute value - equation E.8127 1 Complete the table of values below : x − 3 − 2 − 1 0 1 2 3 x +1 2 · x − 1 2 Among the values studied in the previous question, are there solutions to the equation : ⏐ ⏐ x + 1 ⏐ ⏐ = ⏐ ⏐ 2 x − 1 ⏐ ⏐ E.323 Solve the following equations : a ⏐ ⏐ 2 − x ⏐ ⏐ = 2.5 b ⏐ ⏐ x + 100 ⏐ ⏐ = 1 c 3 × ⏐ ⏐ x + 2 ⏐ ⏐ = 1 d ⏐ ⏐ 2 − x ⏐ ⏐ × 2 = 1 e 3 × ⏐ ⏐ x + 5 ⏐ ⏐ + 3 = 9 f 2 × ⏐ ⏐ 2 − x ⏐ ⏐ + 4 = 1 E.339 Solve the following equations : a ⏐ ⏐ 2 x − 1 ⏐ ⏐ = ⏐ ⏐ − x + 1 ⏐ ⏐ b ⏐ ⏐ 3 x − 1 ⏐ ⏐ = ⏐ ⏐ 3 x + 1 ⏐ ⏐ c ⏐ ⏐ x − 2 ⏐ ⏐ = 5 d ⏐ ⏐ x + 2 ⏐ ⏐ = 6 E.335 Solve the following equations : a ⏐ ⏐ 2 x − 2 ⏐ ⏐ = ⏐ ⏐ x + 3 ⏐ ⏐ b ⏐ ⏐ 2 x + 1 ⏐ ⏐ = ⏐ ⏐ 3 x + 3 ⏐ ⏐ c ⏐ ⏐ x − 2 ⏐ ⏐ = ⏐ ⏐ x + 3 ⏐ ⏐ d ⏐ ⏐ 5 x + 2 ⏐ ⏐ = ⏐ ⏐ 2 · ( − 2 x + 1) ⏐ ⏐ E.7302 Consider the functions f and g defined on R by the relations : f ( x ) = x − ⏐ ⏐ 2 · x − 1 ⏐ ⏐ ; g ( x ) = x − 2 We denote C f and C g respectively the representative curves of the functions f and g . 1 Using a calculator, determine the abscissas of the points of intersection of the curves C f and C g . 2 a Establish equivalence : f ( x )= g ( x ) ⇐⇒ ⏐ ⏐ 2 · x − 1 ⏐ ⏐ = ⏐ ⏐ 2 ⏐ ⏐ b Deduce the coordinates of the intersection points of the curves C f and C g . 4. Absolute value - a little further E.353 Consider the functions f and g defined by: f ( x ) = 1 2 · ⏐ ⏐ ⏐ x + 1 ⏐ ⏐ ⏐ − x ; g ( x ) = 1 4 · x − 1 2 The plane is provided with an orthonormal ( O ; I ; J ) given below. We denote C f and C g the representative curves of the func-tions f and g and consider the following two intervals : I = −∞ ; − 1 ; J = − 1 ; + ∞ 1 Draw the curve C g . 2 a Determine the expressions of the function f on each of the intervals I and J . b Draw the curve C f on R . 3 The following questions will be dealt with algebraically: a Determine the coordinates of the intersection point of the curves C f and C g on the interval J . b Justify that the curves C f and C g do not admit a point of intersection on the interval I . https://chingmath.fr chapExoCorrec/320 sacados/320 chapExoCorrec/8127 sacados/8127 chapExoCorrec/323 sacados/323 chapExoCorrec/339 sacados/339 chapExoCorrec/335 sacados/335 chapExoCorrec/7302 sacados/7302 chapExoCorrec/353 sacados/353 -4-3-2-12345I-3-2-123JO
-4-3-2-1234I-2-123JO -8-6-4-224681012I-4-22468JO E.1939 The aim of the exercise is to solve the following equation : ( E ) : ⏐ ⏐ x + 1 ⏐ ⏐ + 2 ⏐ ⏐ x − 1 ⏐ ⏐ = 3 Consider the function f defined on R by: f ( x ) = ⏐ ⏐ x + 1 ⏐ ⏐ + 2 ⏐ ⏐ x − 1 ⏐ ⏐ 1 a Simplify the expression for the function f on the interval −∞ ; − 1 . b Solve the equation ( E ) on the interval −∞ ; − 1 . 2 a Simplify the expression of the function f on the in-terval − 1 ; 1 . b Solve the equation ( E ) on the interval − 1 ; 1 . 3 a Simplify the expression for the function f on the interval 1 ; + ∞ . b Solve the equation ( E ) on the interval 1 ; + ∞ . 4 Give the set of solutions of the equation ( E ) . E.303 Solve the following equations : a ⏐ ⏐ x + 3 ⏐ ⏐ − ⏐ ⏐ 2 x + 1 ⏐ ⏐ = 2 b ⏐ ⏐ x ⏐ ⏐ + 3 · ⏐ ⏐ 4 x − 1 ⏐ ⏐ = x + 1 E.302 Consider the functions f and g defined on R by: f ( x ) = ⏐ ⏐ x + 2 ⏐ ⏐ ; g ( x ) = ⏐ ⏐ x ⏐ ⏐ − 1 In the plane provided with an orthonormal reference frame O ; I ; J , note C f and C g the respective representative curves of the functions f and g : 1 a Draw the curve C f in the above reference frame. b Graphically solve the inequation f ( x ) 1 . 2 a Draw the curve C g in the above reference frame. b Graphically solve the inequation g ( x ) 0 . E.324 Consider the inequality ( E ) defined by: ( E ) : ⏐ ⏐ x + 1 ⏐ ⏐ + ⏐ ⏐ x − 1 ⏐ ⏐ 5 Consider the following three intervals : I = −∞ ; − 1 ; J = − 1 ; 1 ; K = 1 ; + ∞ 1 Consider the function f defined on R by: f : x ↦−→ ⏐ ⏐ x + 1 ⏐ ⏐ + ⏐ ⏐ x − 1 ⏐ ⏐ Simplify the expression of the function f on each of the three intervals I , J , and K . 2 a Solve the inequality ( E ) on each of the three inter-vals I , J , and K . b Find the set of solutions to the inequality ( E ) . 5. Absolute value - study of functions E.2301 Consider the function f defined on R by: f ( x ) = − 1 4 · ⏐ ⏐ x + 2 ⏐ ⏐ + 3 4 · ⏐ ⏐ x − 4 ⏐ ⏐ 1 a Simplify each expression according to the interval studied : x −∞ − 2 4 + ∞ ⏐ ⏐ x +2 ⏐ ⏐ ⏐ ⏐ x − 4 ⏐ ⏐ b Give the simplified expression of the function f on each of the following three intervals : I = −∞ ; − 2 ; J = − 2 ; 4 ; K = 4 ; + ∞ 2 The plane is provided with an O ; I ; J orthonormal co-ordinate system given below : 3 From the graphical representation, draw the table of vari-ations of the function f . https://chingmath.fr chapExoCorrec/1939 sacados/1939 chapExoCorrec/303 sacados/303 chapExoCorrec/302 sacados/302 -4-3-2-1234I-2-123JO chapExoCorrec/324 sacados/324 chapExoCorrec/2301 sacados/2301 -8-6-4-224681012I-4-22468JO
-6-4-2246I-4-224JOC1 -6-4-2246I-4-224JOC2 -6-4-20246-224(d1(d2(d3 -6-4-20246-224C1 -6-4-20246-224C2 -6-4-20246-224C3 -4-3-2-1234I-2-123JO -4-3-2-1234I-3-2-123JO E.290 Consider the function f defined on R by the relation: f ( x ) = ⏐ ⏐ x + 1 ⏐ ⏐ − 1 2 · x 1 Simplify the writing of algebraic expressions on intervals −∞ ; − 1 and − 1 ; + ∞ in the table below : x −∞ − 1 + ∞ ⏐ ⏐ x + 1 ⏐ ⏐ f ( x ) 2 In the plane provided with a reference frame O ; I ; J orthonormal, are the two curves C 1 and C 2 . Which of these two curves is the representation of the function f : E.326 Consider the function f defined by the relation: f ( x ) = ⏐ ⏐ ⏐ 1 2 · x + 1 2 ⏐ ⏐ ⏐ − ⏐ ⏐ ⏐ 1 4 · x − 1 2 ⏐ ⏐ ⏐ 1 Simplify algebraic expressions on the three intervals −∞ ; − 1 , − 1 ; 2 and 2 ; + ∞ in the table below : x −∞ − 1 2 + ∞ ⏐ ⏐ ⏐ 1 2 · x + 1 2 ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ 1 4 · x − 1 2 ⏐ ⏐ ⏐ f ( x ) 2 In an orthonormal frame, curves are shown below : a By graphical reading and without justification, give the reduced equations of the straight lines ( d 1 ) , ( d 2 ) and ( d 3 ) . b Among the curves C 1 , C 2 and C 3 , which is the repre-sentation of the function f ? E.430 Consider the function f defined on R by: f : x ↦−→ ⏐ ⏐ ⏐ 1 2 x − 1 ⏐ ⏐ ⏐ − ⏐ ⏐ ⏐ − 1 2 − 1 4 x ⏐ ⏐ ⏐ 1 Represent on a graduated line the sets of solutions of each of the following systems of inequalities: 1 2 x − 1 0 − 1 2 − 1 4 x 0 ; 1 2 x − 1 0 − 1 2 − 1 4 x 0 ; 1 2 x − 1 0 − 1 2 − 1 4 x 0 2 Simplify the expression of the function f on each of the following intervals : I = −∞ ; − 2 ; J = − 2 ; 2 ; K = 2 ; + ∞ 3 Draw the curve C f representative of the function f on the interval − 4 ; 4 : E.1497 Consider the function f defined on R by: f ( x ) = 1 2 · ⏐ ⏐ x + 2 ⏐ ⏐ − 1 2 · ⏐ ⏐ x − 2 ⏐ ⏐ 1 Simplify the expression of the function f on each of the following three intervals : I = −∞ ; − 2 ; J = − 2 ; 2 ; K = 2 ; + ∞ 2 The plane is provided with an orthonormal O ; I ; J given below : 3 From the graphical representation, draw the table of vari-ations of the function f . https://chingmath.fr chapExoCorrec/290 sacados/290 -6-4-2246I-4-224JOC1 -6-4-2246I-4-224JOC2 chapExoCorrec/326 sacados/326 -6-4-20246-224(d1(d2(d3 -6-4-20246-224C1 -6-4-20246-224C2 -6-4-20246-224C3 chapExoCorrec/430 sacados/430 -4-3-2-1234I-2-123JO chapExoCorrec/1497 sacados/1497 -4-3-2-1234I-3-2-123JO
-4-3-2-123456I-3-2-1234JO xx0x0 E.5032 Consider the function f defined on R whose image of a number x is given by the relation: f ( x ) = ⏐ ⏐ ⏐ 3 4 · x − 3 2 ⏐ ⏐ ⏐ − ⏐ ⏐ ⏐ 1 2 · x + 1 2 ⏐ ⏐ ⏐ 1 Determine the image of − 4 and 0 by the function f . 2 Simplify the expression of the function f on each of the three intervals below : I = −∞ ; − 1 ; J = − 1 ; 2 ; K = 2 ; + ∞ 3 The plane is given the reference frame O ; I ; J below. In this frame of reference, draw the curve C f representa-tive of the function f . 4 a Graphically and without justification, give the set of solutions to the equation f ( x )= − 1 . b Algebraically, justify that the equation f ( x )= − 1 ad-mits no solution on the interval −∞ ; − 1 . 6. A little further on - algebraic properties of absolute value E.5015 We wish to establish the following equal-ity for all real numbers x and y : ⏐ ⏐ x × y ⏐ ⏐ = ⏐ ⏐ x ⏐ ⏐ × ⏐ ⏐ y ⏐ ⏐ To do this, reason by case disjunction about the value of x and the value of y . Establish this relationship in each of the following cases : a x ∈ R + et y ∈ R + b x ∈ R + et y ∈ R − c x ∈ R − et y ∈ R + d x ∈ R − et y ∈ R − E.5016 1 For all real numbers x and y , establish the inequality: ⏐ ⏐ x + y ⏐ ⏐ 2 ⏐ ⏐ x ⏐ ⏐ + ⏐ ⏐ y ⏐ ⏐ 2 2 Deduce, for all reals x and y , the following comparison : ⏐ ⏐ x + y ⏐ ⏐ ⏐ ⏐ x ⏐ ⏐ + ⏐ ⏐ y ⏐ ⏐ 7. Unclassified financial years E.331 Let x be a number; the distance from x to zero is denoted by | x | and is called the absolute value of x . 1 Complete the following tables : x | x | 3 − 3 4 ; 2 − 2 ı x | x | 3 − 5 2 + 1 5 − √ 10 4 − ı 2 For x 0 , compare x and | x | ? For x 0 , compare x and | x | ? 3 Complete the following diagram accordingly: https://chingmath.fr chapExoCorrec/5032 sacados/5032 -4-3-2-123456I-3-2-1234JO chapExoCorrec/5015 sacados/5015 chapExoCorrec/5016 sacados/5016 chapExoCorrec/331 sacados/331 xx0x0
-4-3-2-1234I-1234JO -4-3-2-10123456 -5-4-3-2-1012345 |x−y|xyxy E.410 Consider the absolute value function de-noted f : f : x ↦−→ | x | 1 Give the definition set of the function f . 2 a Complete the table of values below : x - 4 - 3 - 2 - 1 - 0.5 0 0.5 1 2 3 4 f ( x ) b Draw the curve C f representative of the function f in the frame below : 3 Using the graphical representation, draw up the table of variations of the function f on its set of definition D f . 4 What geometric property does the representative curve of the absolute value function possess? E.979 Let’s see how we can transform the solu-tion of equations of the following form : | x − a | = b where a; b ∈ R into a problem about distances between two points. We will use the following relation, linking distance and abso-lute value : d ( x ; y ) = | x − y | 1 We wish to solve the equation | x − 2 | = 3 a Translate this equation into a problem about the dis-tances between points of abscissa x and 2 . b Use the graduated line below to find the positions of the point of abscissa x solution to this distance prob-lem. c Check that the positions found verify the original prob-lem. 2 We wish to solve the equation : | x +1 | =2 : a Complete the table below : x | x + 1 | d ( x ; 1) d ( x ; − 1) 2 − 4 0 b Justify that : | x +1 | = d ( x ; − 1) . c Translate the starting equation into a problem about point distances. d Use the graduated line below to solve this problem. e Check that the positions found verify the original prob-lem. E.1937 1 Solve the following equations by translating them in terms of distance and with the help of a graduated line: a | x − 4 | = 3 b | x + 1.5 | = 1 c | 2 x + 1 | = 2 d | x − 2 | = | x + 3 | 2 Solve the following equations algebraically: a | x + 2 | = 2 2 b | 3 × ( x + 1) − 2 | = 5 c | 3 x − 2 | = | 3 − 2 x | E.1938 Solve the following inequalities in terms of distance, and plot the set of solutions on a graduated line: a | x + 2 | 5 b | x − 1 | > 2 E.1911 Translate the following equations or in-equalities in terms of distance, then solve them : a | x − 2 | = 1.5 b | x + 1 | = 1 c | 2 x − 1 | = 3 d | x − 3 | 2 e | x + 4 | 1 f | x − 3 | 1 E.11636 On note d ( x ; y ) la distance séparant, sur une droite graduée, les deux points d’abscisse x et y . Nous allons montrer que l’égalité: d ( x ; y ) = | x − y | 1 Compléter les deux tableaux suivants : x y d ( x ; y ) x − y | x − y | 5 − 2 − 3 7 0 5 8 ; 2 3 ; 4 − 5 7 ; 2 2 Comparer d ( x ; y ) et | x − y | . E.11637 Nous allons simplifier l’écriture de | x − y | en fonction des valeurs de x et de y . 1 Donner la forme simplifiée de l’opposé de x − y ? 2 Compléter les phrases suivantes : x − y 0 lorsque x : : : : : : y x − y 0 lorsque x : : : : : : y . 3 Compléter le diagramme suivant : https://chingmath.fr chapExoCorrec/410 sacados/410 -4-3-2-1234I-1234JO sacados/979 -4-3-2-10123456 -5-4-3-2-1012345 chapExoCorrec/1937 sacados/1937 chapExoCorrec/1938 sacados/1938 chapExoCorrec/1911 sacados/1911 sacados/11636 sacados/11637 |x−y|xyxy