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IB Questionbank Maths SL 1
1. Given that f (x) = 2e3x
, find the inverse function f 1
(x).
Working:
Answer:
.......................................................................
(Total 4 marks)
2. Two functions f and g are defined as follows:
f (x) = cos x, 0 x 2;
g (x) = 2x + 1, x .
Solve the equation (g f)(x) = 0.
Working:
Answer:
......................................................................
(Total 4 marks)
IB Questionbank Maths SL 2
3. The diagram shows three graphs.
y
x
B
A
C
A is part of the graph of y = x.
B is part of the graph of y = 2x.
C is the reflection of graph B in line A.
Write down
(a) the equation of C in the form y =f (x);
(b) the coordinates of the point where C cuts the x-axis.
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 4 marks)
IB Questionbank Maths SL 3
4. Let f(x) = 7 2x and g(x) = x + 3.
(a) Find (g f)(x).
(2)
(b) Write down g1
(x). (1)
(c) Find (f g1
)(5).
(2)
(Total 5 marks)
5. Two functions f, g are defined as follows:
f : x 3x + 5
g : x 2(1 x)
Find
(a) f 1
(2);
(b) (g f )(4).
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 4 marks)
IB Questionbank Maths SL 4
6. Let f (x) = 2x, and g (x) =
2x
x, (x 2).
Find
(a) (g f ) (3);
(b) g1
(5).
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 6 marks)
IB Questionbank Maths SL 5
7. Consider the functions f : x 4(x 1) and g : x 2
6 x.
(a) Find g1
.
(b) Solve the equation ( f g1
) (x) = 4.
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 6 marks)
IB Questionbank Maths SL 6
8. Let f (x) = 2x + 1 and g (x) = 3x2 4.
Find
(a) f 1
(x);
(b) (g f ) (2);
(c) ( f g) (x).
Working:
Answers:
(a) ..
(b) ..
(c) .. (Total 6 marks)
IB Questionbank Maths SL 7
9. The function f is given by f (x) = x2 6x + 13, for x 3.
(a) Write f (x) in the form (x a)2 + b.
(b) Find the inverse function f 1
.
(c) State the domain of f 1
.
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(c) ..................................................................
(Total 6 marks)
IB Questionbank Maths SL 8
10. (a) The diagram shows part of the graph of the function f (x) = . px
q The curve passes
through the point A (3, 10). The line (CD) is an asymptote.
15
10
5
-5
-10
-15
C
A
D
y
x15105051015
Find the value of
(i) p;
(ii) q.
IB Questionbank Maths SL 9
(b) The graph of f (x) is transformed as shown in the following diagram. The point A is
transformed to A (3, 10).
y
x
15
15
10
10
5
50
5
5
10
10
15
15
A
C
D
IB Questionbank Maths SL 10
Give a full geometric description of the transformation.
Working:
Answers:
(a) (i) ...........................................................
(ii) ...........................................................
(b) ..................................................................
..................................................................
(Total 6 marks)
11. Let f (x) = ex
, and g (x) = x
x
1, x 1. Find
(a) f 1
(x);
(b) (g f ) (x).
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 6 marks)
IB Questionbank Maths SL 11
12. Let f (x) = x3 4 and g (x) = 2x.
(a) Find (g f ) (2).
(b) Find f 1
(x). (Total 6 marks)
13. Consider the functions f (x) = 2x and g (x) = 3
1
x, x 3.
(a) Calculate (f g) (4).
(b) Find g1
(x).
(c) Write down the domain of g1
.
Answers:
Working:
(a) .....................................................
(b) .....................................................
(c) ..................................................... (Total 6 marks)
IB Questionbank Maths SL 12
14. The function f is defined by ,9
3)(
2xxf for 3 < x < 3.
(a) On the grid below, sketch the graph of f.
(b) Write down the equation of each vertical asymptote.
(c) Write down the range of the function f. (Total 6 marks)
IB Questionbank Maths SL 13
15. The function f is given by f (x) = e(x11)
8.
(a) Find f 1
(x).
(b) Write down the domain of f l
(x).
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 6 marks)
-
16. Let f (x) = ln (x + 2), x 2 and g (x) =
e(x4)
, x 0.
(a) Write down the x-intercept of the
graph of f.
(b) (i) Write down f (1.999).
(ii) Find the range of f.
(c) Find the coordinates of the point of
intersection of the graphs of f and
g. (Total 6 marks)
IB Questionbank Maths SL 14
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 6 marks)
17. Let g (x) = 3x 2, h (x) = 4
5
x
x, x 4.
(a) Find an expression for (h g) (x). Simplify your answer.
(b) Solve the equation (h g) (x) = 0. (Total 6 marks)
18. The functions f (x) and g (x) are defined by f (x) = ex and g (x) = ln (1+ 2x).
(a) Write down f 1
(x).
(b) (i) Find ( f g) (x).
(ii) Find ( f g)1
(x). (Total 6 marks)
IB Questionbank Maths SL 15
19. Let f (x) = 22
3
qx
xp
, where p, q
+.
Part of the graph of f, including the asymptotes, is shown below.
(a) The equations of the asymptotes are x =1, x = 1, y = 2. Write down the value of
(i) p;
(ii) q. (2)
(b) Let R be the region bounded by the graph of f, the x-axis, and the y-axis.
(i) Find the negative x-intercept of f.
(ii) Hence find the volume obtained when R is revolved through 360 about the x-axis. (7)
(c) (i) Show that f (x) =
222
1
13
x
x.
(ii) Hence, show that there are no maximum or minimum points on the graph of f. (8)
IB Questionbank Maths SL 16
(d) Let g (x) = f (x). Let A be the area of the region enclosed by the graph of g and the
x-axis, between x = 0 and x = a, where a 0. Given that A = 2, find the value of a. (7)
(Total 24 marks)
20. Let f (x) = 4x , x 4 and g (x) = x2, x .
(a) Find (g f ) (3).
(b) Find f 1
(x).
(c) Write down the domain of f 1
. (Total 6 marks)
21. The functions f and g are defined by f : x 3x, g : x x + 2.
(a) Find an expression for (f g)(x).
(2)
(b) Find f1
(18) + g1
(18). (4)
(Total 6 marks)
IB Questionbank Maths SL 17
22. The function f is defined by f(x) = 29
3
x, for 3 < x < 3.
(a) On the grid below, sketch the graph of f.
(2)
(b) Write down the equation of each vertical asymptote. (2)
(c) Write down the range of the function f. (2)
(Total 6 marks)
23. Consider f(x) = 5x .
(a) Find
(i) f(11);
(ii) f(86);
(iii) f(5). (3)
IB Questionbank Maths SL 18
(b) Find the values of x for which f is undefined. (2)
(c) Let g(x) = x2. Find (g f)(x).
(2)
(Total 7 marks)
24. Let f (x) = ln (x + 5) + ln 2, for x 5.
(a) Find f 1
(x). (4)
Let g (x) = ex.
(b) Find (g f) (x), giving your answer in the form ax + b, where a, b, . (3)
(Total 7 marks)
25. Let f(x) = 3x, g(x) = 2x 5 and h(x) = (f g)(x).
(a) Find h(x). (2)
(b) Find h1
(x). (3)
(Total 5 marks)
IB Questionbank Maths SL 19
26. Let f be the function given by f(x) = e0.5x
, 0 x 3.5. The diagram shows the graph of f.
(a) On the same diagram, sketch the graph of f1
. (3)
(b) Write down the range of f1
. (1)
(c) Find f1
(x). (3)
(Total 7 marks)
27. Let f(x) = k log2 x.
(a) Given that f1
(1) = 8, find the value of k. (3)
IB Questionbank Maths SL 20
(b) Find f1
3
2.
(4)
(Total 7 marks)
28. Let f(x) = log3 x , for x > 0.
(a) Show that f1
(x) = 32x
. (2)
(b) Write down the range of f1
. (1)
Let g(x) = log3 x, for x > 0.
(c) Find the value of (f 1
g)(2), giving your answer as an integer.
(4)
(Total 7 marks)
29. Let f(x) = cos 2x and g(x) = 2x2 1.
(a) Find
2
f .
(2)
(b) Find (g f)
2
.
(2)
(c) Given that (g f)(x) can be written as cos (kx), find the value of k, k .
(3)
(Total 7 marks)
IB Questionbank Maths SL 21
30. Let f(x) = 2x3 + 3 and g(x) = e
3x 2.
(a) (i) Find g(0).
(ii) Find (f g)(0).
(5)
(b) Find f1
(x). (3)
(Total 8 marks)
31. Let f(x) = x2 and g(x) = 2x 3.
(a) Find g1
(x). (2)
(b) Find (f g)(4).
(3)
(Total 5 marks)