1. The diagram shows three graphs.

B
y
A

A is part of the graph of y = x.

x
B is part of the graph of y = 2 .

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.

2
2. The diagrams show how the graph of y = x is transformed to the graph of y = f(x) in three steps.

For each diagram give the equation of the curve.

y y

(a)
1

0 x 0 x
y= x2 1

(b ) (c ) 7
y

0 1 x 0 1 x

3. The diagram shows the graph of y = f(x), with the x-axis as an asymptote.

1
 y

B (5 , 4 )

A (– 5 , – 4 )

(a) On the same axes, draw the graph of y =f(x + 2) – 3, indicating the coordinates of the images
of the points A and B.

(b) Write down the equation of the asymptote to the graph of y = f(x + 2) – 3.

4. The following diagram shows the graph of y = f (x). It has minimum and maximum points at
1
1,
(0, 0) and ( 2 ).
y

3 .5

2 .5

1 .5

0 .5

–2 –1 0 1 2 3 x
– 0 .5

–1

– 1 .5

–2

– 2 .5

3
y  f ( x – 1) 
(a) On the same diagram, draw the graph of 2.

(b) What are the coordinates of the minimum and maximum points of
3
y  f ( x – 1) 
2?

(Total 4 marks)

2
 2 2
5. The diagram shows parts of the graphs of y = x and y = 5 – 3(x – 4) .

y = x 2

6 2
y = 5 – 3 (x – 4 )

x
–2 0 2 4 6

2 2
The graph of y = x may be transformed into the graph of y = 5 – 3(x – 4) by these
transformations.

A reflection in the line y = 0 followed by

a vertical stretch with scale factor k followed by
a horizontal translation of p units followed by
a vertical translation of q units.

Write down the value of

(a) k;

(b) p;

(c) q.

6. The sketch shows part of the graph of y = f(x) which passes through the points A(–1, 3), B(0, 2),
C(l, 0), D(2, 1) and E(3, 5).

6
E
5

4
A
3
B
2
D
1
C
–4 –3 –2 –1 0 1 2 3 4 5

–1

–2

3
 A second function is defined by g(x) = 2f(x – 1).

(a) Calculate g(0), g(1), g(2) and g(3).

(b) On the same axes, sketch the graph of the function g(x).

q
.
7. (a) The diagram shows part of the graph of the function f(x) = x – p The curve passes through
the point A (3, 10). The line (CD) is an asymptote.
y
C
15

10 A

–15 –10 –5 0 5 10 15 x

–5

–10

–15
D

Find the value of

(i) p;

(ii) q.

(b) The graph of f(x) is transformed as shown in the following diagram. The point A is
transformed to A (3, –10).

4
 y
C
15

10

–15 –10 –5 0 5 10 15 x

–5

–10
A

–15
D

Give a full geometric description of the transformation.

8. Let f(x) = 2x + 1.

(a) On the grid below draw the graph of f(x) for 0  x  2.

(b) Let g(x) = f(x +3) –2. On the grid below draw the graph of g(x) for –3  x  –1.

y
6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 x

–1

–2

–3

–4

–5

–6

5
 2
9. The quadratic function f is defined by f(x) = 3x – 12x + 11.
2
(a) Write f in the form f(x) = 3(x – h) – k.

(b) The graph of f is translated 3 units in the positive x-direction and 5 units in the positive y-
direction. Find the function g for the translated graph, giving your answer in the form g(x) =
2
3(x – p) + q.

10. The graph of y = f(x) is shown in the diagram.

y
2

–2 –1 0 1 2 3 4 5 6 7 8 x
–1

–2

(a) On each of the following diagrams draw the required graph,

(i) y = 2f(x);
y
2

–2 –1 0 1 2 3 4 5 6 7 8 x
–1

–2

(ii) y = f(x – 3).

y
2

–2 –1 0 1 2 3 4 5 6 7 8 x
–1

–2

(b) The point A (3, –1) is on the graph of f. The point A is the corresponding point on the graph

6
 of y = –f(x) + 1. Find the coordinates of A.

x
11. Let f(x) = x, and g(x) = 2 . Solve the equation

–1
(f o g)(x) = 0.25.

12. Two functions f, g are defined as follows:

f : x  3x + 5
g : x  2(1 – x)

Find

–1
(a) f (2);

(b) (g o f)(–4).
3 –l
13. Given functions f : x  x + 1 and g : x  x , find the function (f ° g) .

14. The function f is defined by

3
f :xa 3 – 2x , x .
2
–1
Evaluate f (5) .
3x –1
15. Given that f(x) = 2e , find the inverse function f (x).

x
x
16. Let f(x) = 2 , and g(x) = x – 2 , (x  2).

Find

(a) (g ° f) (3);

–1
(b) g (5).

6–x
17. Consider the functions f : x  4 (x – 1) and g : x  2 .

–1
(a) Find g .
–1
(b) Solve the equation (f ° g ) (x) = 4.

x
and g(x) = 1  x , x  –1. Find
–x,
18. Let f(x) = e

–1
(a) f (x)

(b) (g ° f)(x).

7
 x2 – 1
2
19. The function f is defined for x  0 by f(x) = x  1 .

–1
Find an expression for f (x).

+
20. The set of all real numbers Runder addition is a group ( , +), and the set of all positive real
+ +
numbers under multiplication is a group ( , ×). Let f denote the mapping of ( , +) to ( , ×)
x
given by f(x) = 3 .

+
(a) Show that f is an isomorphism of ( , +) onto ( , ×).
(6)

–1
(b) Find an expression for f .
2
21. The function f is given by f(x) = x – 6x + 13, for x  3.

2
(a) Write f(x) in the form (x – a) + b.

–1
(b) Find the inverse function f .

–1
(c) State the domain of f .

2
22. Let f(x) = 2x + 1 and g(x) = 3x – 4.

Find
–1
(a) f (x);

(b) (g  f )(–2);

(c) (f  g)(x).

23. The functions f and g are defined by f :  3 x, g : x  x  2 .

(a) Find an expression for (f  g) (x).

–l –l
(b) Show that f (18) + g (18) = 22.

x
24. The functions f and g are defined by f : x  e , g : x  x + 2.

–1 –1
(a) Calculate f (3) × g (3).

–1
(b) Show that (f ° g) (3) = ln 3 – 2.

(x–11)
25. The function f is given by f(x) = e –8.
–1
(a) Find f (x).
–l
(b) Write down the domain of f (x).
2
26. The function f is defined for x > 2 by f(x) = ln x + ln (x – 2) – ln (x – 4).

(a) Express f(x) in the form ln(x/(x+a)).

8
 –1
(b) Find an expression for f (x).