EXERCISES ON FUNCTIONS
EXERCISE 1
Are the following curves representing functions?
a) b)
c) d)
e) f)
EXERCISE 2
What is the set of definition of the function ?
a) 𝑓(𝑥) = √1 − 𝑥 2
𝑥 2 −3𝑥+2
b) 𝑓(𝑥) = √ 1
1− 2
𝑥
EXERCISE 3
2
(1−𝑥 2 )
Let 𝑓(𝑥) = .
1+𝑥 2
1) Give the set of definition of 𝑓.
2) Prove that 𝑓 is positive over ℝ.
3) Solve 𝑓(𝑥) ≤ 1.
1
EXERCISE 4
The trajectory of a ball is given by :
𝑓(𝑡) = −5𝑡 2 + 10𝑡 + 15
Where 𝑡 represents the time, in seconds, and 𝑓(𝑡) represents the height of the ball in meters over the
interval [0; 3].
1) Interpret 𝑓(0) and 𝑓(3).
2) What is the maximal height?
3) When is the height equal to 15m ?
4) Solve 𝑓(𝑡) ≥ 18. Interpret the result.
EXERCISE 5
Let 𝑓 and 𝑔 be two functions such that :
1+2𝑥
𝑓(𝑥) = 1+4𝑥
and
1−4𝑥
g(𝑥) = 1−2𝑥.
1) Find the set of definition of 𝑓 and 𝑔.
2) Calculate 𝑓(10−7 ) and 𝑔(10−7 ).
3) Calculate 𝑓(𝑥) − 𝑔(𝑥).
4) Solve : 𝑓(𝑥) − 𝑔(𝑥) > 0.
5) Deduce the sign of : 𝑓(10−7 ) − 𝑔(10−7 ). Conclude.
EXERCISE 6
2𝑥 2 3
Let 𝑓 be the function defined by: 𝑓(𝑥) = 𝑥 2 −1 − 𝑥 2 +𝑥−2.
1) Factorise the following polynomials: 𝑥 2 − 1 and 𝑥 2 + 𝑥 − 2.
2) Determine the set of definition of 𝑓.
3)
2𝑥 2 3
a) Determine the least common denominator of the algebraic fractions and
𝑥 2 −1 𝑥 2 +𝑥−2
𝑔(𝑥)
then write 𝑓(𝑥) as ℎ(𝑥) where ℎ(𝑥) is the least common denominator.
b) Determine a simple root of 𝑔(𝑥).
c) Simplify 𝑓(𝑥) and solve the equation 𝑓(𝑥) = 0.
EXERCISE 7
Let 𝑓 be a numerical function defined by: 𝑓(𝑥) = |2𝑥 + 3| − |3𝑥 − 2|.
1) Rewrite this function without the absolute value.
2) Draw the curve representing the function 𝑓 in an orthonormal reference (O; ⃗i, ⃗j) (unit :0 , 5 cm).
3) Using this graph, solve the inequality |2𝑥 + 3| − |3𝑥 − 2| ≤ 0.
EXERCISE 8
We consider the function 𝑓 defined by 𝑓(𝑥) = 𝑥(1 − 𝑥) over ℝ.
1 1 2
1) Prove that 𝑓(𝑥) = 4 − (𝑥 − 2) .
1
2) Prove that 𝑓(𝑥) ≤ 4, for any x ∈ ℝ.
1
3) Deduce that 𝑓 has a maximum at 𝑥 = .
2
4) Study the monotony of 𝑓.
2
EXERCISE 9
Let 𝑥 and 𝑦 be two positive numbers.
1) Expand the expression (√𝑥 − √𝑦)(𝑥 + 𝑦 + √𝑥𝑦).
2) Deduce the monotony of the function defined by 𝑔(𝑥) = 𝑥 √𝑥 over ℝ+ .
3) Compare the numbers 11 + 30 and 6√6−√5
6−5√5
.
√
EXERCICSE 10
1 1
Let 𝑓(𝑥) = 𝑥 + 𝑥 and 𝑔(𝑥) = 𝑥 − 𝑥.
1) Prove that 𝑓 is increasing over [1; +∞[ and decreasing over ]0; 1].
2) Study the monotony of 𝑔 over ]0; +∞[.
3) Compare 𝑓 and 𝑔 over ]0; +∞[.
EXERCISE 11
𝑥
Let 𝑓(𝑥) = √1 + 𝑥 and 𝑔(𝑥) = 1 + 2.
1) Prove that 𝑓(𝑥) ≥ 0 and 𝑔(𝑥) ≥ 0 for any 𝑥 ∈ [−1; +∞[.
2) Calculate (𝑓(𝑥))2 and (𝑔(𝑥))2 .
3) Prove that (𝑓(𝑥))2 ≤ (𝑔(𝑥))2 for any 𝑥 ∈ [−1; +∞[.
4) Compare 𝑓 and 𝑔.
EXERCISE 12
1 1
Let 𝑓(𝑥) = and g(𝑥) = .
1+𝑥 4 1+𝑥 2
1) Calculate 𝑓(𝑥) − 𝑔(𝑥).
2) Determine the intervals where 𝑓(𝑥) > 𝑔(𝑥).
EXERCISE 13
Study the parity of the function 𝑓.
1
1) 𝑓(𝑥) = 𝑥 + 𝑥
1
2) 𝑓(𝑥) = 𝑥 2 +
𝑥
1
3) 𝑓(𝑥) = 𝑥+ 2
𝑥
1
4) 𝑓(𝑥) = 𝑥 2 + 𝑥 2.
EXERCISE 14
We consider the function 𝑓 defined by 𝑓(𝑥) = 𝑥 2 − 1 and a function 𝑔. Calculate (𝑓 ∘ 𝑔)(𝑥).
1) 𝑔(𝑥) = √1 − 𝑥 ;
1
2) 𝑔(𝑥) = 1 − 𝑥.
EXERCISE 15
Give a decomposition of the function 𝑓 defined by : 𝑓(𝑥) = (𝑥 − 3)2 + 2 which facilitates to conclude
about its variation over I = ]−∞; 3] and conclude about its variation.
3
EXERCISE 16
Let 𝒞f be the graphical representation of 𝑓.
Give the set of definition and the graphical representation of the function 𝑔 in an orthonormal reference
(O; ⃗i, ⃗j) (unit : 1 cm).
1) g(𝑥) = −𝑓(𝑥) ;
2) 𝑔(𝑥) = |𝑓(𝑥)| ;
3) 𝑔(𝑥) = 𝑓(𝑥) + 1 ;
4) 𝑔(𝑥) = 𝑓(𝑥 + 1).
EXERCISE 17
We consider the function 𝑓 defined over ℝ by:
𝑓(𝑥) = 𝑥(𝑥 − 2).
1) Study the parity of 𝑓.
2) Prove that 𝑓(𝑥) = (𝑥 − 1)2 − 1.
3) Prove that 𝑓 is minored by −1.
4) Draw the curve 𝒞𝑓 which represents the function 𝑓 in the interval [−1; 3] in an orthonormal
reference (O; ⃗i, ⃗j) (unit : 0,5 cm).
EXERCISE 18
We consider a function f defined by: 𝑓(𝑥) = 𝑥√4 − 𝑥 2 .
1) Determine the domain of definition 𝐷𝑓 of the function 𝑓.
2) Study the parity of the function 𝑓.
3) Prove that 𝑓 is bounded above by 2 over 𝐷𝑓 . (We can determine the sign of 𝑓(𝑥)2 − 4).
EXERCISE 19
𝑥
Let 𝑓(𝑥) = 1+|𝑥|.
1) Study the parity of the function 𝑓.
2) Prove that ∀x ∈ ℝ, 𝑓(𝑥) < 1.
EXERCISE 20
Let 𝑓 be the function defined by 𝑓(𝑥) = 𝑥 4 .
1) Prove that 𝑥 4 − 𝑦 4 = (𝑥 − 𝑦)(𝑥 + 𝑦)(𝑥 2 + 𝑦 2 ).
2) Prove that 𝑓 is strictly increasing over [0; +∞[.
3) Study the parity of 𝑓. Deduce the monotony of 𝑓 over ]−∞; 0].
4) Draw the graph of 𝑓 in an orthogonal reference (O; ⃗i, ⃗j). The graphical units are 1 cm on the x-
axis and 0,25 cm on the y-axis.
4
EXERCISE 21
1 𝑖𝑓 𝑥 > 0 0 𝑖𝑓 𝑥 ≥ 0
We consider 𝑓 and g two functions such that : f(𝑥) = { and 𝑔(𝑥) = { .
0 𝑖𝑓 𝑥 ≤ 0 −1 𝑖𝑓 𝑥 < 0
1) Graph 𝑓 and 𝑔 in an orthonormal reference (O; ⃗i, ⃗j) (unit : 0,5 cm).
2) Calculate 𝑓 𝑔.
EXERCISE 22
1) Prove that √𝑎 + √𝑏 > √𝑎 + 𝑏, ∀(𝑎, 𝑏) ∈ (ℝ∗+ )2 .
2) Let 𝑓(𝑥) = 2𝑥 2 − 1 and 𝑔(𝑥) = 4𝑥 3 − 3𝑥.
Calculate (𝑓 ∘ 𝑔)(𝑥) then (𝑔 ∘ 𝑓)(𝑥).
Conclude.
EXERCISE 23
1) Let 𝑓(𝑥) = −3𝑥 2 + 5𝑥 − 1.
Find an axis of symmetry of 𝒞𝑓 .
2𝑥−1
2) Let 𝑓(𝑥) = 𝑥+1 .
Prove that A(−1; 2) is a center of symmetry of 𝒞𝑓 .
𝑥 2 −𝑥−2
3) Let 𝑓(𝑥) = 𝑥 2 −𝑥+1.
1
Prove that the straight line of equation 𝑥 = 2 is an axis of symmetry of 𝒞𝑓 .
EXERCISE 24
Let f be a function defined by:
−2 𝑥 ∈ [−7; −5]
2𝑥 + 8 𝑥 ∈ ]−5; −1]
𝑓(𝑥) = 5 7 .
−2𝑥 + 2 𝑥 ∈ ]−1; 1]
{ 𝑥 𝑥 ∈ ]1; 6]
1) Give the table of variation of the function 𝑓.
2) Calculate the images of -2, 0 and 5.
3) Draw the graph of 𝑓 in an orthonormal reference (O; ⃗i, ⃗j) (unit : 1 cm).
2
4) Solve graphically: 𝑓(𝑥) = 0, 𝑓(𝑥) = 1, 𝑓(𝑥) = 4, 𝑓(𝑥) = −𝑥 + 2 and 𝑓(𝑥) ≥ − 3 𝑥.
EXERCISE 25
The function 𝑓 is defined by its curve 𝒞𝑓 .
Give the set of definition of 𝑓 .
Give the table of variations of 𝑓 .
1) 2)
5
3) 4) 5)
EXERCISE 26
Calculate the coordinates of the point(s) of intersection of the curves 𝐶𝑓 and 𝐶𝑔 .
1) 𝑓(𝑥) = 𝑥 2 + 4𝑥 + 1 ; 2) 𝑓(𝑥) = 𝑥 2 + 4𝑥 + 2 ;
2 𝐶𝑔 is the y-axis ;
g(𝑥) = 2𝑥 + 4𝑥 − 3 ;
2
3) 𝑓(𝑥) = 𝑥 + 𝑥 + 1 ; 4) 𝑓(𝑥) = 𝑥 2 + 4𝑥 + 2 ;
𝐶𝑔 is the 𝑥-axis ; 𝑔(𝑥) = 𝑥 2 + 6𝑥 − 1 ;
−2𝑥+3
5) 𝑓(𝑥) = − ; 6) 𝑓(𝑥) = 𝑥(−𝑥 2 + 16)(𝑥 + 2) ;
2𝑥+1
𝐶 𝑖𝑠 𝑡ℎ𝑒 y-axis ; 𝐶𝑔 is the 𝑥-axis .
𝑔
EXERCISE 27
Let be the functions 𝑓 and .
Study the position of the curve 𝒞𝑓 with respect to the curve 𝒞𝑔 .
2𝑥+3
1) 𝑓(𝑥) = 𝑥−1 2) 𝑓(𝑥) = 4 − 𝑥 2
𝑥+3 𝑔(𝑥) = (2𝑥 − 4)(𝑥 + 3)
𝑔(𝑥) =
𝑥−1
EXERCISE 28
Let us consider the following numerical functions 𝑓, 𝑔 and ℎ defined by:
−2𝑥−6
𝑓(𝑥) = −𝑥 2 + 2𝑥 − 2, 𝑔(𝑥) = 2𝑥+1 and ℎ(𝑥) = √𝑥 − 2.
1) Determine 𝐷𝑓 , 𝐷𝑔 and 𝐷ℎ respectively the domains of definition of 𝑓, 𝑔 and ℎ.
1
2) a) Show that, for every 𝑥 ∈ ℝ, we have: 𝑓(𝑥) = −(𝑥 − 1)2 − 1, and for every 𝑥 ∈ ℝ\ {− 2}:
5
𝑔(𝑥) = −1 − 2𝑥+1.
1
b) Show that the line of equation 𝑥 = 1 is an axis of symmetry of 𝐶𝑓 and the point 𝐼(− 2 , −1) is a
point of symmetry of 𝐶𝑔 .
c) Give then the variations of 𝑓, 𝑔 and ℎ.
3) a) Prove that, for every 𝑥 ∈ 𝐷𝑔 , we have:
𝑓(𝑥) = 𝑔(𝑥) ⇔ (𝑥 − 2)(−2𝑥 2 − 𝑥 − 2) = 0
b) Determine then the intersection of the curves 𝐶𝑓 and 𝐶𝑔 .
4) a) How can you deduce the curve 𝐶ℎ from the curve of the function 𝑥 ⟼ √𝑥?
b) Draw 𝐶𝑓 , 𝐶𝑔 and 𝐶ℎ by taking the unit = 1cm. (your graph should take into account the results
of previous questions.)
c) Solve graphically the inequation 𝑓(𝑥) ≥ 𝑔(𝑥).
d) Using the monotony of ℎ, prove that, for every 2 ≤ 𝑥 ≤ 3, we have 0 ≤ ℎ(𝑥) ≤ 1.
6
5) Let 𝑡 be the numerical function defined by 𝑡(𝑥) = −𝑥 + 2√𝑥 − 2.
a) Prove that (𝑓 ∘ ℎ)(𝑥) = 𝑡(𝑥), for every 𝑥 ∈ [2; +∞[.
b) Deduce from the previous questions the variations of 𝑡 over [2; 3] and [3; +∞[.
EXERCISE 29
Let 𝑓 be a function over ℝ, such that for every 𝑥 ∈ ℝ:
1 + 𝑓(𝑥)
𝑓(𝑥 + 1) =
1 − 𝑓(𝑥)
1) Calculate 𝑓(𝑥 + 2), 𝑓(𝑥 + 3) and 𝑓(𝑥 + 4).
2) What can we deduce from that for the function 𝑓?
EXERCISE 30
Let 𝑓 and 𝑔 two functions defined over ℝ. What one can say about the parity of 𝑔 ∘ 𝑓 in the following
cases?
1) 𝑓: 𝑥 ↦ |𝑥| and 𝑔 is any function ;
2) 𝑔: 𝑥 ↦ |𝑥| and 𝑓 is any function ;
3) 𝑓 is even and 𝑔 is any function ;
4) 𝑓 is odd and 𝑔 is any function ;
5) 𝑓 and 𝑔 are odd ;
6) 𝑓 is odd and 𝑔 is even.
EXERCISE 31
4
Let us consider the function 𝑓 defined over ℝ by: 𝑓(𝑥) = − 3 𝑥 2 + 8𝑥.
1) Determine the images by 𝑓 of 2, 3 and 7.
2) Determine the antecedents by 𝑓 of 0.
3) Study the sign of 𝑓(𝑥).
4
4) Verify that for every real 𝑥, one has: 𝑓(𝑥) = − 3 ((𝑥 − 3)2 − 9).
5) Prove that the function 𝑓 is strictly increasing over ]−∞; 3]and strictly decreasing over [3; +∞[.
6) Deduce from 5 that 𝑓 has a maximum. For which value of 𝑥 this maximum is attained?
This maximum has to be calculated.
7) Draw carefully the curve representing 𝑓 in an orthonormal reference (O; ⃗i, ⃗j) (unit : 0,5 cm).
EXERCISE 32
In an orthonormal reference (O; ⃗i, ⃗j) (unit : 0,5 cm), graph the function 𝑓.
Find graphically the sets of solutions (S1 ) and (S2 ) such that:
𝑓(𝑥) = 0 (S1 ) and 𝑓(𝑥) > 0 (S2 ).
1) 𝑓(𝑥) = 3𝑥 + 2
2) 𝑓(𝑥) = 1 − 𝑥
3) 𝑓(𝑥) = 𝑥 2 + 1
7
EXERCISE 33
The curve below represents a function 𝑓 defined over [−3; 3] :
1) Give the table of variations of the function 𝑓.
2) Solve graphically the following equations:
a) 𝑓(𝑥) = 1 b) 𝑓(𝑥) = 0
c) 𝑓(𝑥) = −1 d) 𝑓(𝑥) = 2
3) Determine the sign of 𝑓(𝑥).
1 1
4) Solve graphically the equation 𝑓(𝑥) = 2 𝑥, and then the inequality 𝑓(𝑥) ≤ 2 𝑥.
EXERCISE 34
We consider a function 𝑓 defined over ℝ by: 𝑓(𝑥) = 3(𝑥 − 1)2 + 2.
1) Prove that 𝑓 is strictly increasing over [1; +∞[.
2) Prove that 𝑓 is bounded below by 2 over ℝ.
3) Solve the equation 𝑓(𝑥) = 5.
4) Determine two functions 𝑔 and ℎ such that: 𝑓 = 𝑔 ∘ ℎ.
EXERCISE 35
The aim of this exercise is to compare the two following numbers:
A = 1,0000002 and B = √1,0000004.
𝑥
Let 𝑓 and 𝑔 be two functions defined by: 𝑓(𝑥) = √1 + 𝑥 and 𝑔(𝑥) = 1 + .
2
1)
a) Find the domains of definition 𝐷𝑓 and 𝐷𝑔 of the functions 𝑓 and 𝑔 respectively.
b) Verify that 𝑓(4 × 10−7 ) = 𝐵. What can you say about 𝑔(4 × 10−7 )?
2) In order to compare the numbers A and B, we would like to compare the functions 𝑓 and 𝑔.
a) Prove that 𝑓(𝑥) ≥ 0 and 𝑔(𝑥) > 0 for every x ∈ [−1; +∞[.
2 2
b) Calculate (𝑓(𝑥)) and (𝑔(𝑥)) .
2 2
c) Prove that (𝑓(𝑥)) < (𝑔(𝑥)) for every x ∈ [−1; +∞[\{0}.
d) Deduce that 𝑓(𝑥) < 𝑔(𝑥) for every x ∈ [−1; +∞[\{0}.
e) Conclude.
