Pure Mathematics Year 2 (A Level) Unit Test 3: Functions and Modelling
1 , x∈ℝ
a Sketch the graph of y = f(x), labelling its vertex and any points of intersection
with the coordinate axes. (5 marks)
b Find the coordinates of the points of intersection of and
(5 marks)
2 The functions p and q are defined by and
a Given that pq(x) = qp(x), show that (4 marks)
b Explain why has no real solutions. (2 marks)
3 The functions f and g are defined by , x∈ℝ and ,
x∈ℝ, x > −1
a Find fg(x) and state its range. (4 marks)
b Solve fg(x) = 85 (3 marks)
4 The function g(x) is defined by , x∈ℝ, x > 4. Find g−1(x) and
state its domain and range. (6 marks)
5 The diagram shows the graph of h(x).
Figure 1
The points A(−4, 3) and B(2, −6) are turning points on the graph and C(0, −5) is the
y-intercept. Sketch on separate diagrams, the graphs of
a y = |f(x)| (3 marks)
b y = f(|x|) (3 marks)
c y = 2f(x + 3) (3 marks)
Where possible, label clearly the transformations of the points A, B and C on your
new diagrams and give their coordinates.
© Pearson Education Ltd 2017. Copying permitted for purchasing institution only. This material is not copyright free. 1
1 , x∈ℝ
a Sketch the graph of y = f(x), labelling its vertex and any points of intersection
with the coordinate axes. (5 marks)
b Find the coordinates of the points of intersection of and
(5 marks)
2 The functions p and q are defined by and
a Given that pq(x) = qp(x), show that (4 marks)
b Explain why has no real solutions. (2 marks)
3 The functions f and g are defined by , x∈ℝ and ,
x∈ℝ, x > −1
a Find fg(x) and state its range. (4 marks)
b Solve fg(x) = 85 (3 marks)
4 The function g(x) is defined by , x∈ℝ, x > 4. Find g−1(x) and
state its domain and range. (6 marks)
5 The diagram shows the graph of h(x).
Figure 1
The points A(−4, 3) and B(2, −6) are turning points on the graph and C(0, −5) is the
y-intercept. Sketch on separate diagrams, the graphs of
a y = |f(x)| (3 marks)
b y = f(|x|) (3 marks)
c y = 2f(x + 3) (3 marks)
Where possible, label clearly the transformations of the points A, B and C on your
new diagrams and give their coordinates.
© Pearson Education Ltd 2017. Copying permitted for purchasing institution only. This material is not copyright free. 1
