This document contains 9 multi-part math problems involving functions: 1) Define functions f and g and find their domains, ranges, and solve equations. 2) Explain the domain of a function g. 3) Define functions h and k, find compositions and domains/ranges. 4) Write a function f in a certain form and find its range and domain. 5) Sketch a graph and find stationary points. 6) Solve an equation involving functions g and h. 7) Find ranges of functions f and g, and find domains, solutions, and evaluations. 8) Show relations between functions g and h, sketch graphs, and find domains/ranges. 9

1. The function f is defined by for .1
Write down the range of f.(a)
[2]
Find and state its domain and range.(b)
[4]
The function g is defined by for .
Solve .(c)
[3]
[Total: 9]
1

2. The domain of is such that exists. Explain why is a suitable domain for g(x).2
[1]
[Total: 1]
The functions h and k are defined by
for ,
for .
3
Find hk(10).(a)
[2]
Find , stating its domain and range.(b)
[5]
[Total: 7]
A function f is such that for .4
Show that can be written in the form , where a and b are integers.(a)
[2]
2

3. Write down the range of f.(b)
[1]
FInd and state its domain.(c)
[3]
[Total: 6]
On the axes below, sketch the graph of showing the coordinates of the points
where the graph meets the axes.
(a)5
[3]
3

4. Find the coordinates of the stationary point on the curve .(b)
[2]
Find the values of k such that the equation has only 2 solutions.(c)
[2]
[Total: 7]
4

5. Functions g and h are such that, for x ∈,
and .
Solve .
6
[4]
[Total: 4]
It is given that for ,
for .
7
Write down the range of f and of g.(a)
[2]
5

6. Find , stating its domain.(b)
[3]
Find the exact solution of .(c)
[4]
6

7. Evaluate .(d)
[2]
[Total: 11]
The functions g and hg are defined, for , by8
,
.
7

8. Show that .(a)
[2]
(b) The diagram shows the graph of . Given that g and h are inverse functions, sketch, on the same
diagram, the graph of . Give the coordinates of any point where your graph meets the coordinate
axes. [2]
8

9. State the domain of h.(c)
[1]
State the range of h.(d)
[1]
[Total: 6]
9
O x2
4
y
The diagram shows the graph of passing through and touching the x-axis at . Given
that the graph of is a straight line, write down the two possible expressions for .
[2]
[Total: 2]
9