Page 20 - Focus SPM 2022 - Additional Mathematics

Additional Mathematics SPM Chapter 1 Functions

D Determining a related function given 20
composite function and another 1
function A function f is defined as f : x → , x ≠ 0.
x
2
3
4
1. When a composite function and one of its (a) Express f (x), f (x) and f (x) in the simplest
functions is given, the remaining function can forms.
41
be determined. (b) Hence, deduce the outcomes of f (x) and f (x).
42
Solution
19 (a) f (x) = f [f(x)]
2
Given g(x) = 2x – 1. Find the function h if 1
(a) gh(x) = 7 – 6x = f 3 4
x
3
(b) hg(x) = , x ≠ 0 1
2x = Form 4
Solution 1 x
(a) gh(x) = 7 – 6x = x
g[h(x)] = 7 – 6x
3
2
2h(x) – 1 = 7 – 6x f (x) = f [f (x)]
2 h(x) = 8 – 6x = f(x)
h(x) = 4 – 3x = 1
3 x
(b) hg(x) = 4 2 2
f (x) = f [f (x)]

2x 3Pelangi Sdn Bhd. All Rights Reserved.
2
h[g(x)] = 3 = f (x)
2x = x
h(2x – 1) = 3
2x Alternative Method
4
3
Let y = 2x – 1, f (x) = f [f (x)]
1
2 x = y + 1 = f 1 2
x = y + 1 x
2 = 1
3 1
h(y) = x
2 1 y + 1 2 = x
2
4
2
3 (b) Note that f (x) = x and f (x) = x
=

Therefore, h(x) = x + 1 , x ≠ –1 The same applies to f(x) = and f (x) =
1
1
y + 1
Penerbitan By deduction, f (x) = x when n is even and
3
x
x
n
1
Try Questions 8 – 13 in ‘Try This! 1.2’
f (x) = when n is odd.
n
x
1
Therefore, f (x) = and f (x) = x.
42
41
x
E Solving problems involving composite
functions
1. We can combine a function f with itself to form Try Questions 14 – 16 in ‘Try This! 1.2’
the composite function ff which is also written as
f . 21
2
2. The combination of f and f will form the The suggested retail price of a new hybrid car is RMp.
2
composite function f f or f f or f . A car company has advertised in the newspapers that
2
3
2
the car manufacturer offers a rebate of RM5 000 while
3. We can continue using this method to generate the car company itself gives a discount of 10%.
composite functions such as f , f and so on.
4
5
17