Page 19 - Focus SPM 2022 - Additional Mathematics

Additional Mathematics SPM Chapter 1 Functions
Solution Given g(y) = z,
(a) fg(x) = f [g(x)] g(y) = y 2
= f(2x + 3) g(m) = m = k
2
= (2x + 3) 2 g(5) = 5 = k
2
= 4x + 12x + 9 k = 5
2
2
= 25
(b) gf(x) = g[f(x)] (b) gf(x) = g(2x + 1)
= g(x ) 2
2
= 2x + 3 = (2x + 1)
2
= 4x + 4x + 1
2
(c) f (x) = ff(x) Try Question 5 in ‘Try This! 1.2’
2
= f [f(x)]
= f(x )
2
= (x ) C Determining the image of composite
2 2
Form 4
= x 4 functions given the object, and vice
SPM Tips versa
2
f (x) means ff(x), which means the function f acts 17
on the object x twice. f (x) ≠ [f(x)] 2
2
In the diagram on the x f y h z
right, the function f maps 9
2
(d) g (x) = gg(x) x to y and the function h
= g[g(x)] maps y to z.
= g(2x + 3) Find 5
= 2(2x + 3) + 3 (a) f(3), 3
= 4x + 6 + 3 (b) hf(3).

= 4x + 9
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Solution
Try Questions 3 – 4 in ‘Try This! 1.2’ (a) f(3) = 5
(b) hf(3) = h[f(3)]
= h(5)
16 = 9
Try Question 6 in ‘Try This! 1.2’
x f y g z
k
18
m
Given the functions f : x → (x – 1) – 2, x . –1 and
2
2
g : x → 2x + 8 , x . –1. Determine the composite
In the diagram above, the function f maps x to y such x + 1
that y = 2x + 1 and the function g maps y to z such function fg(5).
that z = y . Find
2
(a) the values of m and k, Solution
(b) the expression for gf(x). g(5) = 2(5) + 8 Function g acts on
5 + 1 5 first.
Solution = 3
(a) Given f(x) = y, fg(5) = f(3)
f(x) = 2x + 1 = (3 – 1) – 2
2
f(2) = 2(2) + 1 = 5 = 2
m = 5
Try Question 7 in ‘Try This! 1.2’

16