Page 28 - Focus SPM 2022 - Additional Mathematics

Additional Mathematics SPM Chapter 1 Functions
6. When constructing a composite function f (a) (b)
with f or f with f, we will obtain an identity x f y a g b
–1
–1
function, x. The function f and its inverse f can p
–1
“cancel out” each other. a 4 x q
b 5 y r
34 c 6 z s t
A function f is defined as f(x) = 3x. Set X Set Y Set A Set B
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–1
(a) Find the inverse function, f .
(b) Hence, verify the truth that the composite (c) h = {(3, 12), (4, 16), (5, 20), (6, 24)}
–1
functions f [f (x)] and f [f(x)] are equal to the
–1
identity function, x. 4. Determine whether the graph of functions of f, g and
h in the following diagrams have inverse functions.
Solution Explain your answers. Form 4
(a) The function f multiplies each input with 3. To (a) y
reverse the operation that defined f(x), we divide 3
each input with 3. Hence, the inverse function of f 2
x
f(x) is f (x) = —. 1
–1
3 –3 –2 –1 0 1 2 3 x
x
(b) f [f (x)] = f 1 2 –1
–1
3
–2
x
= 3 1 2 = x –3
3
(b) y
Likewise, f [f(x)] = f (3x)
–1
–1
3x 5
= 4
3 g 3
= x 2
Therefore, f [f (x)] = f [f(x)] = x 1 x
–1
–1
–3 –2 –1 0 1 2 3
Try Question 16 in ‘Try This! 1.3’ –1
–2
Try This! 1.3 (c) y
3
1. In the arrow diagram on the x f y 2
right, the function f maps x to 10– 1 h
1
y. Determine 2 0 x
(a) f (8), 2 –1 –1 1 2 3 4 5 6 7 8 9
–1
(b) the value of p if –3 –2
f (p) = 2.
–1
8 –3
2. In the diagram on g 5. Determine whether the following graphs of functions
the right, the function f x f y z of f and g have inverse functions. Explain your
maps y to x while the 2 answers.
function g maps y to z. (a) y
Determine 0 2
(a) f (–2), f
–1
–1
(b) gf (–2), –2 1
(c) g (2), –2 –1 0 1 2 3 x
–1
(d) fg (2). –1
–1
–2
3. Determine whether each of the following functions –3
has an inverse function. Give reasons for your –4
answers. –5
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