Exercise 1:
Let f be the function defined on by
1) a) Calculate the image of by f , the result is given in the form with a and b as decimals.
b) Repeat the previous question with .
2) Determine the antecedents of 5 by f.
3) a)Prove that for any real x we have .
b) Derive the factorized expression for f(x).
c) Determine the antecedents of 0 by f
Exercise 2:
Consider the function f defined on by:
.
1. Does f(3) = 1 ?
2. Are the images of 2 and 0 by f equal?
3. Determine the image of by f.
4. Determine the antecedents of 0 by f.
Exercise 3:
Here is the representative curve of a function f defined on .
By graphical reading, determine:
a) the image of 1 by f.
b) the image of 0 by f.
c) the antecedent(s) of 1 by f.
d) the antecedent(s) of 3 by f.
Exercise 4:
Here is the representative curve of a function g defined on [2 ; 3].
By graphical reading, determine:
a) g(0).
b) the images of 1 and 2 by g.
c) the possible antecedents of 1; 1 and 5.
Exercise 5:
Here is the representative curve of a function f defined on [0;7].
Estimate the solutions of the following equations.
a) f(x)= 2 b) f(x)=0 c) f(x) = – 1 d) f(x) = 1.
Exercise 6:
Here is the representative curve of a function g defined on [5 ; 5].
Estimate the solutions of the equations.
a) g(x) = 2.
b) g(x)= 3.
c) g(x) =4.
d) g(x) = 1.
Exercise 7:
Here is the representative curve of a function k defined on [3 ; 4].
Estimate the solutions of the following equations and inequalities.
a) k(x) =1 b) k(x) = 0 c) k(x)> 1
d) k(x) < 0 e) k(x) 2 f) k(x) 2
Exercise 8:
Here is the representative curve of a function h defined on [5 ; 5].
Estimate the solutions of the following inequalities.
a) h(x) 0
b) h(x) < 4
c) h(x) < 2
d) h(x) > 2
Exercise 9:
Here are the representative curves of a function f
and a function g defined on [2;3].
Solve equations and inequalities graphically.
a) g(x) = f(x)
b) g(x) f(x)
c) f(x) < 3
d) g(x) < 2
e) f(x) 2
Exercise 10:
For each of the curves below, say whether it appears to be the representative curve of an even function, an odd function, or a function that is neither even nor odd.
Exercise 11:
For each of the functions whose expressions are given below,
try to establish the largest possible set of definitions.
Exercise 12:
Consider the representative curves of the inverse function, denoted f,
and the affine function g defined on R by g(x) = 2x + 1.
They are plotted in the marker below.
1. Locate the curves associated with the two functions.
2. Solve graphically the equation .
3. a) Expand the expression (2x – 1)(x+ 1).
b) Find algebraically the results obtained in question 2.
Exercise 13:
Consider a function f whose graphical representation is given on [3;4].
1. Determine the image of 2.
2. Give the value of f(2).
3. Give an approximate value of the antecedents of 5.
4. Solve for f(x) = 4.
5. Solve f(x) < 6.
6. Solve f(x) 8.
Exercise 14:
Let f be the function defined by
1) Determine Df, the definition set of f.
2) Determine the images of 1, 7 and 4.
3) Determine graphically the antecedent(s) of 2.
4) Determine by calculation the antecedent(s) of 1.
5) Graph f.
Exercise 15:
Let the function f be defined by
1) Determine the definition set of f.
2) Determine the images of 0 and 2.
3) Determine the antecedent(s) of 2 and 0.
4) Determine the sign of f.
5) Draw the graphical representation of f.
Exercise 16:
We define the function f in ℝ by
1) Calculate the image of .
2) Determine the antecedent(s) of 0 and .
3) Study the sign of f.
4) Draw the curve Cf.
Exercise 17:
Let f be the function defined on by :
1) Determine the sign of f.
2) Does f have an extremum?
3) Graph f.
Exercise 18:
Let f be the function defined on ]3 ; +∞[ by
1) Study the sign of f.
2) Study the variations of f.
3) Draw the curve Cf.
Exercise 19:
f is the function defined in ℝ by
1) Determine the antecedent(s) of 1, 4 and 0
2) Determine the sign of f.
3) Show that f has an extremum which we will characterize.
4) Study the variations of f.
5) Graph f.
Exercise 20:
Let f be the function defined by
1) Determine Df.
2) Calculate the images of 0 and .
3) Study the variations of f.
4) Graph f.
5) Solve graphically: f (x) = 3.
6) Solve graphically: f (x) > x.
Exercise 21:
Let f : and g : .
1) Study the variations of f. then those of g.
2) Graphically represent these two functions
3) Determine the relative position of Cf and Cg.
Exercise 22:
Determine the direction of variation of the functions below:
1) x (x + 1^{)2} – 5 on ]; 1]
2) x on ]; 1/2]
3) x on ^{R}
4) x 2 + ]2 ; +[
5) x on ^{ R+}
6) x ( x  + 1^{)2 } on ^{R}
7) x 3 + ]1 ; +[
8) x on ]1 ; +[
9) x on ]0; 1]
10) x on [0; 1[
Exercise 23:
Describe the direction of variation of the function defined by the given curve.
Exercise 24:
Describe the direction of variation of the following function.
Exercise 25:
The table below gives the direction of variation of a function f defined on the interval [3;4].
For each statement, say whether it is true or false. Justify your answer.
1.f is increasing on [0;2].
2.f is decreasing on [1;1].
3.f(3)>f(2).
4.
5.For any real number x in the interval [3;2], .
Exercise 26:
f is a function defined on the interval [3;6] such that :
 the maximum of f on [3;6] is equal to 5, it is reached for x=0.
 the minimum of f on [3;6] is equal to 2, it is reached for x=3.
 the antecedents of 0 by f are reached for x=0;3 and 6.
 the maximum of f on [3;1] is equal to 3 and it is reached for x= – 2.
 f(1)=2 and f is increasing on [1;0].
Draw a curve that can represent the function f in this frame.
Exercise 27:
g is a decreasing function on such that g(0)=1 and g(1)=0.
What is the set of real numbers such that :
a)
b) g(x)<1.
Exercise 28:
The questions 1. à 5. Contains three possible answers. For each of these questions, only one of the proposed answers is correct. You are asked to check this answer.
An inaccurate answer removes half of the points allocated to the question. Failure to answer a question earns no points and takes none away. If the total is negative, the score is reduced to 0.
Let f be a function defined on the interval [5;5] whose table of variation is the following:(we suppose that the representative curve of f on [5;5] is obtained “without raising the pencil”)

at one point in two points in three points 

equal to 0 negative less than 2 

and are negative 



and are of opposite signs 
Exercise 29:
A handball player throws a ball in front of her.
After x meters travelled, the height of the ball (in meters) before it touches
the soil is given by: .
How high is the ball after 20 meters?
What can we deduce for the ball?s
a) Show that .
b) What can we say about the sign of ?
c) Deduce the maximum height reached by the ball.
Exercise 30:
Consider a rectangle of length 7 and width 5.
We draw inside this one a cross of width x
variable as shown below.
We are interested in the area of the blue cross.
1. To which interval does x belong ?
2. Express the area of the blue cross as a function of x.
3. With the calculator, draw the table of values of with a step of 1.
Exercise 31:
The price of unleaded gasoline is 1 euro per liter.
Marius wants to fill up his car. He plans to put x liters in
its empty tank which can contain 40 liters.
The station in which it is used does not deliver less than 5 liters.
Consider the function P that associates to each value of x
the price paid by Marius.
1. According to the context of the exercise, to which interval does x belong?
2. What is the set of definition of the function P?
3. Determine the algebraic expression of the function P.
Exercise 32:
ABCD is a rectangle such that AB 10 cm and BC = 8 cm.
N is a moving point on the segment [BC].
We note x the length in centimeters M and P are the respective points of [AB] and [CD] such that
AM = BN = CP = x.
The goal of this exercise is to determine where to place N on the segment [BC] so that the area of the yellow surface, the sum of the areas of the triangles BMN and CNP, is maximal.
1. Justify that .
2. Express BM as a function of x.
3. Express CN as a function of x.
4. Show that the area of the triangle BMN is equal to
5. Let f be the function that associates the total area of the yellow surface with the length x.
Check that we have .
6.a) Show that .
b) Deduce the solution to the problem.
Exercise 33:
Associate each function with its table of variations
among the following.
a) f defined on by f(x) = 2x+ 4.
b) g defined on by g(x) = .
c) h defined on by h(x) = .
d) k defined on by k(x) = .
Exercise 34:
A function f has the following properties:
– it is defined on [3 ; 5] ;
– it is increasing on [3 ; I] ;
– it is decreasing on [1 ; 4] ;
– it is increasing on [4 ; 5] ;
– on the interval [3 ; 4], its maximum is 6 ;
– on the interval [1 ; 5], its minimum is 3 ;
– the image of 3 is I ;
– 5 is an antecedent of 7.
Draw the table of variations of this function.
Exercise 35:
Here is the table of variations of a function f.
Choose the curve corresponding to this table.
Exercise 36:
Here is the table of variations of a function f.
Compare if possible the following numbers and justify.
1. a) f(2) and f(4).
b) f( 2) and f( 1).
2. Solve .
3. It is further known that .
Solve and .
Exercise 37:
Consider a square of side 15 cm.
In each corner, we cut the same square to obtain a pattern of a box without a lid.
A. A special case
1. Construct the pattern of a box by choosing BM = 3 cm.
2.calculate its volume.
3. Can we make a box knowing that BM = 8 cm ?
Explain.
B. A function
We pose BM= x and we call V the function which associates to x the volume of the box without lid.
1. Determine an expression for the function V.
2. What is the defining set of V?
3. Using a calculator or a software, draw the representative curve of the function V.
4. For what values of x is the volume greater than or equal to 100 ?
5. Can the volume of this box exceed 1 dL?
If so, give the dimensions of a box verifying this condition.
If not, explain why.
Exercise 38:
Consider a rectangle ABCD of dimensions AB = 6 cm and BC = 8 cm.
On the side [AB], we place any point M.
Then consider the points N on [BC], P on [CD] and Q on [DA] such that
AM = BN = CP = DQ.
Let AM = x. We call f the function that x associates the value of the area of MNPQ.
1. Can AM take the value 7 ?
What is the definition set of f?
2. Show that.
3. Using a calculator or software, draw the curve
Representative of f. Adjust the display window.
For what value(s) of x is the area of MNPQ greater than or equal to 24 cm² ?
Exercise 39:
g is a function whose table of variations is known.
a) Give the direction of variation of the function g on the interval [2 ; 5].
b) Determine which number is greater than g(3) and g(4).
2. On the model of the previous question, compare g (1) and g(l ,5).
3. Same question for g (2) and g (O).
Exercise 40:
Copy and complete the proposed table of variations from the following graph.
Exercise 41:
Copy and complete the proposed table of variations from the following graph:
Exercise 42:
Copy and complete the proposed table of variations from the following graph.
Exercise 43:
Determine a and b so that the table below is a table of values of a function h defined by
on .
2. Is the function h even ? odd?
3. Determine the antecedents of 7 by h.
Exercise 44:
Consider the function f defined on by f(x) = 2x+ 5.
1. Determine the antecedent(s) of 2 by f.
2. Write an algorithm or program that:
– requests a value from the user;
– calculates and displays the antecedent(s) of b by the
function f.
Exercise 45:
1. Using the calculator, copy and complete the table of values of the function h defined
on [2 ; 2] by h(x)=(3x+1)(5x).
2. Determine all antecedents of 0 by h.
Skills to learn about numerical functions:
 Know the definition of functions;
 Calculate an image or an antecedent;
 Set up a table of values or variations of functions;
 Establish a table of signs;
 Know how to use the graphical representation of numerical functions.
These exercises are in accordance with the officialnational education programs.
In addition, you can consult the course on numerical functions in second grade.
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