Page 9 - Top Class Additional Mathematics Tg 4

Additional Mathematics Form 4 Chapter 1 Functions

11. For each pair of the following functions, find PL 3
Bagi setiap pasangan fungsi yang berikut, cari
(i) fg(2) and gf(4), / fg(2) dan gf(4),
(ii) the value of x when fg(x) = 5. / nilai x apabila fg(x) = 5.

Example (a) f : x → 3x + 2, g : x → 2 – x 2
f : x → 5x, g : x → 4 – 2x
2
(i) fg(x) = f(2 – x )
(i) fg(x) = f[g(x)] gf(x) = g[f(x)] = 3(2 – x ) + 2
2
= f(4 – 2x) = g(5x) = 6 – 3x + 2
2
= 5(4 – 2x) = 4 – 2(5x) = 8 – 3x 2
= 20 – 10x = 4 – 10x
fg(2) = 8 – 3(2)
2
fg(2) = 20 – 10(2) gf(4) = 4 – 10(4) = –4
= 0 = –36
gf(x) = g(3x + 2)
= 2 – (3x + 2)
2
Alternative Method
= 2 – (9x + 12x + 4)
2
g(2) = 4 – 2(2) f(4) = 5(4) = –9x – 12x – 2
2
= 0 = 20
2
fg(2) = f(0) gf(4) = g(20) gf(4) = –9(4) – 12(4) – 2
= 5(0) = 4 – 2(20) = –194
= 0 = –36
(ii) fg(x) = 5
2
8 – 3x = 5
(ii) fg(x) = 5 3x = 3
2
20 – 10x = 5 x = 1
2
10x = 15 x = 1 or –1
x = 1.5
12. Find the function g based on the functions f and fg given. PL 4
Cari fungsi g berdasarkan fungsi f dan fg yang diberikan.

Example (a) f(x) = x + 4, fg(x) = 3x + 9 (b) f(x) = 3x – 6, fg(x) = 5x + 8
f(x) = x + 3, fg(x) = 2x – 5
f [g(x)] = 3x + 9 f [g(x)] = 5x + 8
g(x) + 4 = 3x + 9 3g(x) – 6 = 5x + 8
f [g(x)] = 2x – 5
g(x) + 3 = 2x – 5 g(x) = 3x + 5 3g(x) = 5x + 14
5x + 14
g(x) = 2x – 8 g(x) =
∴ g : x → 3x + 5 3
∴ g : x → 2x – 8 ∴ g : x → 5x + 14
3

13. Find the function f based on the functions g and fg given. PL 4
Cari fungsi f berdasarkan fungsi g dan fg yang diberikan.

Example (a) g(x) = 2x + 3, fg(x) = 7x – 4 (b) g(x) = x – 9, fg(x) = 3x – 5
2
2
g(x) = x + 2, fg(x) = 4x – 9
2
f[g(x)] = 7x – 4 f[g(x)] = 3x – 5
2
f(2x + 3) = 7x – 4 f(x – 9) = 3x – 5
2
f[g(x)] = 4x – 9
2
f(x + 2) = 4x – 9 Let y = 2x + 3 Let y = x – 9
2
Let y = x + 2 x = y – 3 x = y + 9
2
x = √y + 9
x = y – 2
2
So, f(y) = 4(y – 2) – 9 So, f(y) = 7  y – 3  – 4 So, f(y) = 3(√y + 9) – 5
2
= 3(y + 9) – 5

= 4y – 8 – 9 = 3y + 27 – 5
= 4y – 17 = 7y – 29
2 = 3y + 22
Substitute y with x, f(x) = 4x – 17 ∴ f : x → 3x + 22
∴ f : x → 4x – 17 ∴ f : x → 7x – 29
2
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