Cambridge IGCSE and O Level Additional Mathematics
9 The function g is defined by g (x ) = 1 − 2x 2 for x 0.
The function h is defined by h(x ) = 3x − 1 for x 0.
Solve the equation gh (x ) = −3 giving your answer(s) as exact value(s).
↦ x 2 for x ∈ ℝ. g : x ↦ x + 2 for x ∈ ℝ.
10 The function f is defined by f : x
The function g is defined by
Express each of the following as a composite function, using only f and g:
a x
x ↦ x + 4 d x ↦ x 2 + 2 c x ↦ x4 ↦ (x + 2 )2 b
11 The functions f and g are defined for x > 0 by f : x
↦
x + 3 and g : x
↦
Express in terms of f and g:
a x
↦
x + 3 b x ↦ x + 6 c x ↦
12 Given the functions f (x ) = x and g (x ) = a Find the domain and range of g.
x +3
x−5 , 2x + 1
b Solve the equation g (x ) = 0. c Find the domain and range of fg.
2.4 Modulus functions REMINDER
• The modulus (or absolute value) of a number is the magnitude of the number without a sign attached.
14
• The modulus of x, written as |x |, is defined as x if x > 0 x = 0 if x = 0 −x if x < 0
• The statement |x |, = k, where ⩾ 0, means that x = k or x = –k.
worked example 4 a
4x + 3 = x + 18 b
2x 2 − 9 = 7
Answers a
4x + 3 = x + 18 4x + 3 = x + 18 or 4x + 3 = −x − 18 3x = 15
5x = −21
x =5
x =−
Solution is : x = 5 or − b
2x 2 − 7 = 9
2x 2 − 7 = 9 2x 2 = 16 x 2 = 8
21 5
21 5
or 2x 2 − 7 = −9
x = ±2 2 Solution is : x = ±2 2
2x 2 = −2 x 2 = −1
x