Functions Hicham Ibrahim

I- Calculate the following limits

II- Calculate the limit of the function f at infinity

III- Find the domain of definition of f then the limit of f at the endpoints of its domain

IV-

V- Prove that ( D ) is an asymptote of the curve of f

VI- The following questions are independent.
2𝑥−1+√𝑥 2 +4
a- Calculate the following limit : lim ( )
x   3𝑥−1
b- Show that the line (D) of equation y = x + 2 is an oblique asymptote at +∞ of the curve (C ) of
function f(x) = 2x - √𝑥 2 − 4𝑥 .
3x 2  2 if x  4
c- Given: f(x) =  . Find a and b knowing that f is continuous and differentiable over IR.
ax  b if x  4
d- Let f be a function differentiable over IR such that f(x2) + f(x) = 2x. Calculate f '(0) & f ’(1).
3sin x  5
e- Consider the function f defined by: f(x) = .
x2  7
8 2
Show that:  f (x)  2 . And deduce the limit of f at +  .
x 7
2
x 7
f- Study the concavity of the curve of f(x) = x3 – 3x2 + 12x – 5.
g- Given the function 𝑔(𝑥) = 𝑥 3 − 3𝑥 2 + 1

a- Study the variation of g.

b- Prove that 𝑔(𝑥) = 0 has one unique root 𝛼 in ]0.6 ; 0.7[ and give an approximate of 𝛼.
c- Prove that 𝑔(𝑥) has an inflection point to be determined.

h- Find the equation of the tangent to the curve of f at the point A(2;f(2)) such that f(x) =x 3x  2 .
i- Find the equation of the tangent to the curve of f at B(0.5 ; f(0.5)) such that𝑓(𝑥) = √2𝑥 − 1 .
x2  x 1  2
j- Given: f  x   . Study the continuity of f at the point of abscissa xo = 1.
x2
x2  1
k- Let f  x   2 and (C) its representative curve.
x   m  3 x  6  m
1. Determine m such that x = 2 is a vertical asymptote to (C). And deduce the other asymptotes.
2. Let g  x   f  x  . Determine m so that g(x) is defined for any real number x.
l- Let f be the function defined by f(x) = x3 + bx2 + cx + d and let ( C ) be its representative
curve.Determine b , c and d such That this function admits an extremum at point ( 1 ; 2) and its curve
admits at point of abscissa 0 a tangent parallel to a line of equation (d): y = - 9x + 11

VII-A- Given the function g(x) = x3+x-2

1- Verify that g(x) = (x-1)(x2+x+2)
2- Deduce that g(x) has the same sign as (x-1)
𝑥−1
B- given the function f defined on R\{0} by 𝑓(𝑥) = 𝑥 − 𝑥 2 and let ( C ) be its curve in an orthonormal
system
1- Find a vertical asymptote of ( C ).
2- i- Calculate the limit of f at infinity.
ii- Show that the line ( D ) of equation y=x is an asymptote of ( C ) at ±∞.
iii- Study the relative position of ( D ) & ( C ).
𝑔(𝑥)
3- i- Show that 𝑓 ′ (𝑥) = 𝑥 3 and set up the table of variations of f.
ii- Find the equation of the tangent ( T ) to ( C ) which is parallel to ( D ).
4- Show that f(x) = 0 admits a unique root α in the interval ] - 1,4 ; - 1,3 [.
5- Draw ( C ) , ( T ) & the asymptotes
VIII-Find f’(xA) such that ( T ) is tangent to ( C ) at A and draw the table of variation of f.

IX-
Given the curve of a function Find the curve of:g(x)=f’(x);h(x)/h’=f;L(x)=|f(x)|;K(x)=f(-x);J(x)=-
f f(x);E(x)=f(|x|)
X- Shown below is the table of variations of a function f :

x – –1 0 1 +

f '(x)
+ + 0 – –
+ –1 +
f (x)

1 – – 1

Designate by (C) the representative curve of f in an orthonormal system.

Part A
1) Determine the domain of definition of f.
2) Give the equations of the asymptotes of (C).
3) What is the number of solutions of the equation f(x) = 3 ?
4) Solve the inequality f(x) < 0.
5) Compare f(2) and f(3), with justification.
6) Write an equation of the tangent to (C) at the point A(0 ;-1).
7) Draw the curve (C)
Part B

ax 2  1
In this part we let f ( x )  .
x b
2

1) Use the table of variations of f to determine the values of a and b.

2) Solve the equation f(x) = 3.
Part C Given the following functions:
1
𝑔(𝑥) = (𝑓(𝑥))3 ; ℎ(𝑥) = (𝑓(𝑥))2 ; 𝐿(𝑥) = √𝑓(𝑥); 𝑎𝑛𝑑 𝐽(𝑥) =
𝑓(𝑥)

1) Determine the domain of definition of g.

2) Calculate g’.
3) Draw the table of variations of g.
4) Redo the same questions for the functions h , L , and J.

VII-. Let f be a function defined over IR by: f(x) =  x  x 2  8 .

1- Show that f is decreasing over IR and draw the table of variations of f.
2- Find the limits of f and show that the representative curve (C) of f admits, as asymptotes, the
straight-line of equation y = 2x and the axis of abscissas.
3- Draw (C) in an orthonormal system. (Take 2 cm as a graphical unit).
 1
XI-Given the function f ( x)  x  1  defined on R\{3}, and let (C) be the representative curve in an
x3

orthonormal reference (O, i, j ).

1-Calculate the limit of f(x) at  ,,3 and 3  ,then deduce an asymptote to (c ).

2-Prove that the line (d) y=x+1 is an asymptote to(C) and study the relative position of (d) and ( C ).

3-Study the intersections of (C ) and the axes of the system.

( x  4)( x  2)
4-Prove that f ' ( x)  , and draw the table of variations of f(x).
( x  3) 2
5-Write the equation of the tangent (T) to (C ) at the point of abscissa x=2,5.
6-Draw the curve (C ) and the asymptotes.
7-Solve graphically: i) f(x)<0 ;ii ) f(x)>6.
8-Draw the curve of the function g(x)=f(/x/).

XII-The following table is the table of variations of a function f defined over  {1}.
 
Denote by (C) the representative curve of f in an orthonormal system (O ; i , j ).
x  0 1 2 
f ’(x)  0
– –
0

0  
f (x)
  4

A– For each of the following statements, answer True or False and justify.

1 3
1) If x  0 , then f (x)  0 . 2) f  f . 3) f '( 0.5)  f '(0.5) .
2 4
4) The x-axis is tangent to the curve (C) at the origin O.
1
B- In what follows, let 𝑓(𝑥) = 𝑎𝑥 + 𝑏 + 𝑥−𝑐 where a , b & c are three real numbers.

1
1) prove that f (x)  x  1 
x 1
x 2  2x
2) a) Show that f '(x)  .
(x  1) 2
b) Write an equation of (T), the tangent to the curve (C) at its point A with abscissa x  2 .
c) Find the equations of the tangents to ( C ) that are parallel to the second bisector : y = - x
3) Verify that the line (d) with equation y = x +1 is an asymptote to (C).

4) Draw (d) and (C) .

5) Show that the equation f (x)  2 has two distinct solutions.

XIII- Let f be the function defined on  ;1  1;   as f  x    x  2 
1
.
x 1
(C) is the graph of f in an orthonormal system.

1) Determine lim f (x) , lim f (x) lim f (x) and lim f (x) ,and deduce an asymptote    to (C).
x 1 x 1 x   x  
x 1 x 1

2) Show that the line  d  : y = – x + 2 is an asymptote to (C) and study the relative position of (d) and ( C ).

3) Find the coordinates of I the intersection of    & (d) and prove that I is a center of symmetric of ( C ).

x 2  x
4) Show that f  (x)  and set up the table of variations of f.
 x  1
2

5) Draw    ,  d  and (C).

6) Find the equations of the tangents to ( C ) that are parallel to the first bisector : y = x.

8) Prove that y = –2x + 3 is an equation of the line passing through I and joining the two

vertices of (C). then Solve the inequality: f(x) < – 2x + 3.

9). Draw the curve of the following functions: a- g(x)=|f(x)| ; b- h(x)=f( |x| ) ; c- L(x)= - f(x)
2
XIV- Consider the function f defined over ]   ;  2[  ]  2 ; +  [ by f(x) = a x +b +
x2
where a and b are two real numbers (a  0). Denote by ( C ) its representative curve
in an orthonormal system (O; i , j ) .
The table of variations of f is the following:

x  2 +

+ +

f(x)

 

A-
1) Compare, with justification, f(  4) and f(  3).
2) What is the number of solutions of the equation f(x) =  2?
3) Knowing that f(  3) = 0 and that f(0) = 0, calculate a and b .
2
B- In this part, suppose that f(x) =  x  1+ .
x2
1) Verify that the line  Δ  with equation x=  2 and the line (D) with equation y =  x  1 are
asymptotes to (C). and study the relative position of (D) and ( C ).

2) Draw  Δ  , (D) and (C).

3) a- The line (d) with equation y = x intersects the curve (C) in two distinct points. Calculate
 
the coordinates of these points and draw (d) in the same system ( O; i , j ) .
2
b- Using (C) and (d), solve the following inequality :  x  1 +  x.
x2
XV-
The curve (C) drawn below is the graphical representation of a function f.
The straight lines (L) and (D) are the asymptotes of (C).

(C)

(D)

-1
O . 1 3
x

(L)

Use the above graphical representation to:

1) a) Determine the domain of f . and write an equation of the line (D) & (L).
b) Find lim f (x ) , lim f (x ) , lim f (x) and lim f (x) .
x   x  
x 1 x 1
x 1 x 1
2) Find f(0) , f(3) , f (1) and f (3) .
3) Solve : a) f(x) > 0; b) f(x)  1 c) f ( x )  0 d) |f(x)| > 5 e ) f(x) ≥ - 4
(𝑥+2)𝑓(𝑥)
f ) 4 < f(x) ≤ 5 g) [f(x)]2-16=0 h) [f(x)]2-f(x)-20=0 i) ≥0
𝑓′(𝑥)
4) Set up the table of variations of f.
b
5) The function f is given by f(x) = ax +1 + . Show that a = – 1, b = – 4 & c=1.
xc
6) .Find an equation of the tangent to (C) at the point of abscissa 0.
7) Discuss according to the values of the real m the number of solution of the equation
f(x)=m.
8) Draw the curve of the following functions:a- g(x)=|f(x)|; b- h(x)=f( |x| ); c- L(x)= f(-x)
 XVI
Given below the table of variations of a function f that is defined on IR.
(C ) is the representative curve of f in an orthonormal system of axes.

x  2 
f '(x)  0 +
1 1

f (x)
1

4

A- Answer with true or false, justifying your answer:

1) (C) has an asymptote parallel to the axis of abscissas. 2) f (0)  f (3) .

3) f (0)  f (1) . 4) (C) cuts the axis of abscissas at one point only.

5) The tangent to (C) at the point of abscissa 2 has an equation y =1.

x 2  4x  3
B- Suppose that f, defined on IR, is given by f (x)  .
x 2  4x  8
1) Calculate the coordinates of the points of intersection of (C) with the axis of abscissas.

2) Write an equation of the tangent to (C) at the point of abscissa 0.

3) Draw (C).
XVII-

XVIII- The following table represents the table of variations of a function f whose representative curve, in an
orthonormal system, is (C).
x  3 1 0 2 3 +
f (x) +    + 

f(x)

2 5  + 3 5 + + 2
1) Find the domain of definition of f.
2) Find the limits of f at the bounds of the domain of definition.
3) Find the equations of the asymptotes of (C). and draw ( C ).
4) What is the number of solutions of the equation: f(x) = 0? Justify.
 XIX- The adjacent curve (C)
represents a function f. y
Using (C),
a- find the following limits
1) lim f ( x ) .
x  2
2) lim f ( x ) . 1
x 
3) lim f ( x ) . x -3 -2 O 1 2 x
x 0 
4) lim f ( x ) .
x 0 
b- Solve :
(𝑥+3)𝑓′(𝑥)
≥0
𝑓(𝑥)
y
c- choose the best answer:
A B C
1) (C) admits 1 asymptote 2 asymptotes 3 asymptotes
2) The equation of the oblique
y=x y=x+1 y=x+2
asymptote is:
3) lim f ( x )   0 
x 
4) The domain of definition of f
]  ;   [ IR* None
is:
5) The equation of the tangent to
x=1 y1=0 y=x
(C) at (2 , 1) is:
An axis of A center of No elements of
6) (C) has
symmetry symmetry symmetry
7) The straight-line y = 2 and (C)
No points 1 point Two points
intersect in
 x 3  5x
XX- Consider the function f defined over IR by: f(x) = . Let (C) the representative curve of f
x2  3
bx
1) Show that f(x) = ax  where a and b are two real numbers to be determined.
x 3
2

2) Show that f is odd. Deduce the element of symmetry of (C).

3) a- Calculate the limits of f at the boundaries of its domain of definition.
b- Determine f (x) and study its sign.
c- Construct the table of variations of f.
4) Find the equation of the straight-line (T), tangent to (C) at O.
5) Draw ( C )
2x 2  x  1
XXI- Consider the function f defined by: f(x) = .
x 1
1) Find Df, the domain of f. & Determine: lim f ( x ) , lim f ( x ) , lim f ( x ) , and lim f ( x ) .
x   x   x 1 x 1

and lim f ( x )  2x  .
f (x)
2) Determine: lim
x   x x  

3) Study the variations of f.and draw ( C )

4) Prove that the representative curve (C) of f has a center of symmetry to be determined.
 XXII- Consider the function f defined, over IR, by: f(x) = x3 + 3x2  2 where (C) is its representative curve
 
in an orthonormal reference O; i, j .
1) Study the variations of f
2) Study the concavity of ( C ) ,Deduce a center of symmetry of ( C )
3) Prove that the equation f(x)=0 admits one unique solution α in [2.73 ; 2.74 ]
2
4) Prove that 𝛼 2 = 3−𝛼
5) Draw (C).
6) Let (L) be the straight-line of equation: y = m where m is a real parameter. Study, according to the values
of m, the number of the points of intersection between (C) and (L).
XXIII- Choose the best answer: (Justify your answer)
A B C
1) The function f(x) = x2  4 is: Even Odd Neither even nor odd

x3  x
2) The function g(x) = is: Even Odd Neither even nor odd
x2
3) The axis of symmetry of the function
x=1 x=0 x = 1
h(x) = x2 + 1 is:
4) let f(5+h) = 4+6h – 3h2+2h5.
y = 4x − 13 y = 6x − 4 y = 6x − 26
then the equation of the tangent to Cf at x=5 is:

5) The domain of definition of the function

3 
2x  3 [4 ; +  [ ]4 ; +  [  2 ;4 
g(x) = is:  
4x

 x2  3  2
 if x  1
6)Given f (x)   x 1 1 1
-1 
ax 2  2x  1 if x  1 2 2
 2
If f is continuous at 1, then a =

7) If f(x) = 2x  3 , then the number 1

2 1
derivative of f at 2 is: 2

XXIV-
The adjacent figure represents
the curve of a function g. y

f is the function such that

g(x) = f (x).

Find the sense of variations of f.

x O x

y

Hicham Ibrahim