Page 23 - Focus SPM 2022 - Additional Mathematics

Additional Mathematics SPM Chapter 1 Functions

23 24
In the following arrow diagram, the function f maps x Determine whether each of these functions have an
to y while the function h maps y to z. inverse function. Give reasons for your answers.
(a) (b)
x f y h z
9 x f y x g y
Penerbitan Pelangi Sdn Bhd. All Rights Reserved.
1 p 1 p
5
2 q 2 q
3
3 r 3 r
Determine Set X Set Y Set X Set Y
(a) h(5),
(b) f (5), (c) h = {(1, 2), (1, 4), (2, 6), (2, 8)}
–1
(c) h (9),
–1
Form 4
(d) hf(3), Solution
(e) f h (9). Function Reason Conclusion
–1 –1
Solution (a) f Each element x in the f is a one-to-one
(a) h(5) = 9 domain is mapped to function, hence
only one element y in f has an inverse
(b) f (5) = 3 the codomain. function.
–1
(c) h (9) = 5
–1
(b) g The inverse of this g is not a one-
(d) hf(3) = h[f(3)] function maps one to-one function,
= h(5) = 9 element y in the hence g does not
(e) f h (9) = f [h (9)] codomain to three have an inverse
–1
–1
–1 –1
= f (5) = 3 elements x in the function.
–1
domain.
Try Question 2 in ‘Try This! 1.3’ (c) h Objects 1 and 2 each h is not a one-
have two images, to-one function,
1 → 2, 1 → 4, 2 → 6, hence h does not
B Making and verifying conjectures 2 → 8. have an inverse
related to the properties of inverse function.
functions
Try Question 3 in ‘Try This! 1.3’
(a) The properties of inverse functions
(I) Only one-to-one functions have inverse 25
functions
Determine whether the graphs of function of f and g
1. A function f which maps set X to set Y has below have inverse functions. Explain your answers.
an inverse function f if f is a one-to-one (a) y
–1
function. 4
2. For a one-to-one function, each element in the 3 f
domain is mapped to only one element in the 2
codomain. The inverse of this function also 1
maps each element in the codomain to only one –1 0 1 2 3 4 5 6 7 x
element in the domain.

20