1. Consider the functions given below.
f(x) = 2x + 3
1
g(x) = x , x ≠ 0
(a) (i) Find (g ○ f)(x) and write down the domain of the function.
(ii) Find (f ○ g)(x) and write down the domain of the function.
(2)
(b) Find the coordinates of the point where the graph of y = f(x) and the graph of
y = (g–1 ○ f ○ g)(x) intersect.
(4)
(Total 6 marks)
2. The diagram below shows the graph of the function y = f(x), defined for all x ,
where b > a > 0.
1
Consider the function g(x) = f ( x a) b .
(a) Find the largest possible domain of the function g.
(2)
IB Questionbank Mathematics Higher Level 3rd edition 1
(b) On the axes below, sketch the graph of y = g(x). On the graph, indicate any
asymptotes and local maxima or minima, and write down their equations and
coordinates.
(6)
(Total 8 marks)
3. The quadratic function f(x) = p + qx – x2 has a maximum value of 5 when x = 3.
(a) Find the value of p and the value of q.
(4)
(b) The graph of f(x) is translated 3 units in the positive direction parallel to the x-
axis. Determine the equation of the new graph.
(2)
(Total 6 marks)
IB Questionbank Mathematics Higher Level 3rd edition 2
4. The diagram shows the graph of y = f(x). The graph has a horizontal asymptote at y = 2.
1
(a) Sketch the graph of y = f ( x) .
(3)
(b) Sketch the graph of y = x f(x).
(3)
(Total 6 marks)
IB Questionbank Mathematics Higher Level 3rd edition 3
5. Shown below are the graphs of y = f(x) and y = g(x).
If (f g)(x) = 3, find all possible values of x.
(Total 4 marks)
IB Questionbank Mathematics Higher Level 3rd edition 4
ax
6. The graph of y = b cx is drawn below.
(a) Find the value of a, the value of b and the value of c.
(4)
IB Questionbank Mathematics Higher Level 3rd edition 5
b cx
(b) Using the values of a, b and c found in part (a), sketch the graph of y = a x
on the axes below, showing clearly all intercepts and asymptotes.
(4)
(Total 8 marks)
2x 3
7. A function f is defined by f(x) = x 1 , x ≠ 1.
(a) Find an expression for f–1(x).
(3)
(b) Solve the equation │f–1(x)│ = 1 + f–1(x).
(3)
(Total 6 marks)
IB Questionbank Mathematics Higher Level 3rd edition 6
4 x2
8. Let f(x) = 4 x.
(a) State the largest possible domain for f.
(2)
(b) Solve the inequality f(x) ≥ 1.
(4)
(Total 6 marks)
9. Find the set of values of x for which │x – 1│>│2x – 1│.
(Total 4 marks)
a
x
10. (a) Let a > 0. Draw the graph of y = 2 for –a ≤ x ≤ a on the grid below.
(2)
IB Questionbank Mathematics Higher Level 3rd edition 7
3x
2
11. Consider the function g, where g(x) = 5 x .
(a) Given that the domain of g is x ≥ a, find the least value of a such that g has an
inverse function.
(1)
(b) On the same set of axes, sketch
(i) the graph of g for this value of a;
(ii) the corresponding inverse, g–1.
(4)
(c) Find an expression for g–1(x).
(3)
(Total 8 marks)
4
, x 2
12. Let f (x) = x 2 and g (x) = x − 1.
If h = g ◦ f, find
(a) h (x);
(2)
(b) h−1 (x), where h−1 is the inverse of h.
(4)
(Total 6 marks)
IB Questionbank Mathematics Higher Level 3rd edition 8
13. The graph of y = f (x) for −2 x 8 is shown.
1
,
On the set of axes provided, sketch the graph of y = f x clearly showing any
asymptotes and indicating the coordinates of any local maxima or minima.
(Total 5 marks)
2x
1
x 1
14. Find all values of x that satisfy the inequality .
(Total 5 marks)
IB Questionbank Mathematics Higher Level 3rd edition 9
IB Questionbank Mathematics Higher Level 3rd edition 10
