# Multiple-choice questions

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```					Essential Mathematical Methods 3 & 4 CAS
Chapter 1 Functions and relations: SAC Revision

Multiple-choice questions
1 For the function with rule f(x) =       25  x , which of the following is the maximal domain?
A (, 25]
B [5, 5]
C (, 25]
D (25, )
E [25, )

7
2 If f is a function for which the rule is f(x) = 8  x, where x is real, the rule for the inverse

function f 1 is:
8
A f 1(x) = + x
7
8
B f 1(x) = 7

2x + 73
C f 1(x) =      4
7
D f 1(x) = 8  x

8
E f 1(x) = 7  x

3 For f: [ 1, 5) R, f(x) = x2, the range is:
A R
B [0, )
C [0, 25)
D [4, 25]
E [0, 5]

1
Essential Mathematical Methods 3 & 4 CAS
Chapter 1 Functions and relations: SAC Revision

4 A function has rule, f(x) = 5x1/3  1, x  R. The rule for the inverse function is:
5
A f 1(x) = x  1

1
B f 1(x) =
5x  1
3

x+1
C f 1(x) = ( 5 )3

x 1
D f 1(x) =    3
5
E f 1(x) = 5x3 + 1

5 (2, 6)  (, 3] =
A (, 2)
B (, 6)
C (2, 3]
D (6, 3)
E (2, 6)

6 (2, 6)  (3, 3] =
A (3, 6)
B (, 6]
C (2, )
D (3, 2)
E (2, 3]

7 Which of the following functions is not one-to-one?
A f(x) = 9  x2, x  0
1
B f(x) = x2  9

C f(x) = 1 9x
D f(x) = x
3
E f(x) =
x

2
Essential Mathematical Methods 3 & 4 CAS
Chapter 1 Functions and relations: SAC Revision

8 The function f: (3, 4]  R, f(x) = x2  3 has range:
A (6, 13)
B [ 3, 13]
C (0, 6)
D [0, 13]
E (3, 6]

9 The range of the function with graph as shown is:

y

(4, 12)

(, 0)                     y = x2  4

4                        x

A R
B (1, 12)
C (0, 3)
D [4, 12)
E [0, 12)

1
10 State the maximal domain of the function with rule f(x) =
x+5
A R\{5}
B R\{0}
C R+
D [5, )
E (5, )

11 The range of the function with rule f(x) = |x  4| + 3 is:
A (4, )
B R
C [3, )
D (4, )
E (1, )

3
Essential Mathematical Methods 3 & 4 CAS
Chapter 1 Functions and relations: SAC Revision

12 For f: (a, b]  R, f(x) = 5  x where a < b the range is:
A (5  a, 5 b)
B (5  a, 5  b]
C (5  b, 5  a)
D (5  b, 5  a]
E [5  b, 5  a)

13 If f(x) = x2 + 1 and g(x) = 2x + 1 then f(g(a)) =
A 4a2 + 4a + 1
B 4a
C 4a2 + 4a + 2
D 4a2 + 1
E 2a2 + 1

2x + 6 if x  2
14 If f (x) =                   then the range of f is:
 x + 2 if x < 2

A (, 10]
B (, 2)
C (, 2]
D [2, )  (2, 0]
E R

1
15 For the function with rule f(x) = x , f(a + 2) + f(a  2) =


A
a2  4

B a


C 2a

1
D 4

2a
E
a 4
2

4
Essential Mathematical Methods 3 & 4 CAS
Chapter 1 Functions and relations: SAC Revision

1 Sketch the graph of f: [–1, 5]  R, f (x) = 2x + 3 and state the range of this function.

2 For the function f: [1, 4]  R, f (x) = 2x2
a state the range of f
b state the rule and domain of the inverse function f –1
c on the one set of axes sketch the graphs of f and f –1

3                2x + 1
3 Find the inverse, f –1, of the function f: R\{– 5 }  R, f (x) = 5x + 3. State the domain and

range of the inverse function.

4 Sketch the graph of f: [3, )  R, f (x) = x – 3 and find the inverse function f –1. Sketch
the graph of f –1 on the same set of axes.

1
5 Sketch the graph of f: R\{0}  R, f(x) = x + 3 and on the same axes sketch the graph of

f –1. State the domain and range of f –1.

6 Let f: [–2, )  R, f(x) = 3 + 4x. Find f –1, stating the rule, domain and range, and sketch
the graph of y = f(x) and y = f –1(x) on the one set of axes.

7 If f(x) = | x | and g(x) = x2  2x + 3 find:
a f(g(1))
b g(f(1))
c f(g(2))
d g(f(2))

8 For f(x) = | x | 5 and g(x) = x 2  find f(g(x)) and g(f(x)) and state the range of the
functions with rule y = f(g(x)) and y = g(f(x))

5
Essential Mathematical Methods 3 & 4 CAS
Chapter 1 Functions and relations: SAC Revision

Extended-response questions
A piece of fencing 240 m long will be used to enclose
three sides of a rectangular field.
The fourth side has a brick wall.
Let l (m) be the length of the field as shown. Let A
(m2) be the area of the field.

1 Express A as a function of l.
2 What is a suitable domain of the function?
3 Use a graph to determine the range of A.

6
Essential Mathematical Methods 3 & 4 CAS
Chapter 1 Functions and relations: SAC Revision

Answers to Chapter 1 Test A

1 A
2 D
3 C
4 C
5 C
6 A
7 B
8 B
9 D
10 E
11 C
12 E
13 C
14 A
15 E

1

(–1, 5)   y

3

0                     3               x
2

(5, –7)

range = [–7, 5]

7
Essential Mathematical Methods 3 & 4 CAS
Chapter 1 Functions and relations: SAC Revision

2 a [2, 32]
x
b f –1(x) =           2 and domain = [2, 32]
c

(4, 32)
y
y = f(x)

(1, 2)                              (32, 4)
–1
y = f (x)
(2, 1)

0                                               x

2              –3x +1               3
3 domain=R\{5 }, f –1(x) = 5x – 2 , range = R\{–5 }

4 f –1(x) = x2 + 3

y
y = f –1(x)

3
y = f(x)

0                                               x
3

1
5 f –1(x) = x – 3 , domain = R \{3}and range = R\{0}

y

y = f(x)

3

0                3 y = f –1(x)             x

8
Essential Mathematical Methods 3 & 4 CAS
Chapter 1 Functions and relations: SAC Revision

6

y
y = f(x)

y = f –1(x)
3       3
–4
0       3                          x
3
(–5, –2)                     –4

(–2, –5)

x–3
f–1(x) =
4
domain = [–5, ), range = [–2,
7 a 2
b 2
c 11
d 3
8 f(g(x)) = | x2– 5| – 5, range = [–5, ); g(f(x)) = x2 – 10 |x | + 20, range = [–5, )

1, 2    A(l) = l(240 – 2l) where 0 < l < 120
3 (0, 7200]

9

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