Page 11 - Pra U STPM 2022 Penggal 1 - Mathematics (T)

Mathematics Term 1 STPM Chapter 1 Functions

Hence, g is the inverse function of f and is written as f . The inverse of a function f can be obtained by
–1
inverting the direction of the arrow in the diagram, such as the one shown below.
1 f

a b

f –1
Domain of f Range of f
–1
–1
(Range of f ) (Domain of f )
Figure 1.6
For the function f : x ↦ 2x + 1, the inverse of f is f : x ↦ 1 (x – 1).
–1
2
1
x f 2x +1 f –1 – (x–1)
x 2
–1 –1 –1 –1
0 1 1 0
1 3 3 1

2 5 5 2
A B B A
Figure 1.7(a) Figure 1.7(b)
The diagram in Figure 1.7(a) shows the function f : x ↦ 2x + 1. The diagram in Figure 1.7(b) shows the inverse
of the function f, which maps each element in the set B to only one element in the set A. This relation is a
one-to-one relation. Hence, the inverse of the function f is also a function, and is called the inverse function
of f.
Now consider the function f below,
2
i.e. f : x ↦ x , x  R
–1
–1
2
The inverse of the function f is f : x ↦ x or f : x ↦ ± x .
x f x 2 x f –1 + – x
–2 4 4 –2
–1 1 –1
1 1 1
2 2
P Q Q P
Figure 1.8(a) Figure 1.8(b)
2
The diagram in Figure 1.8(a) shows the function f : x ↦ x . In the inverse of function f, as shown in Figure
1.8(b), the elements 4 and 1 in the set Q are each mapped to two elements in the set P. Each of the elements
4 and 1 does not have a unique image. Hence, the inverse of the function f is not a function.
Now consider the function f again, where
2
f : x ↦ x , x  R +
The inverse of this function is
–1
f : x ↦  x (positive square root only)

+
2
Hence, the inverse of the function f is also a function, i.e. if f : x ↦ x , x  R 0 , then the inverse function of
+
–1
f is f : x ↦  x, x  R 0 .
8

01a STPM Math T T1.indd 8 3/28/18 4:20 PM