Cambridge International AS and A Level Mathematics Pure Mathematics 1

Functions

118

P1^

4

This result can be used to obtain a sketch of the inverse function without having

to find its equation, provided that the sketch of the original function uses the

same scale on both axes.

Finding the algebraic form of the inverse function

To find the algebraic form of the inverse of a function f(x), you should start by

changing notation and writing it in the form y = ....

Since the graph of the inverse function is the reflection of the graph of the original

function in the line y = x, it follows that you may find its equation by interchanging

y and x in the equation of the original function. You will then need to make y the

subject of your new equation. This procedure is illustrated in Example 4.4.

ExaMPlE 4.4 Find f−^1 (x) when f(x) = 2 x + 1, x ∈.

SOlUTION

The function f(x) is given by y = 2 x + 1

Interchanging x and y gives x = 2 y + 1

Rearranging to make y the subject: y =

x–1

2

So f−^1 (x) =

x− 1

2 , x^ ∈

Sometimes the domain of the function f will not include the whole of . When

any real numbers are excluded from the domain of f, it follows that they will be

excluded from the range of f−^1 , and vice versa.

x

y

A(0, 4)

A(4, 0)

B(1, –1)

B(–1, 1)

C(–4, 2)

C(2, –4)

y = x

Figure 4.8

f

domain of f and

range of f–1

range of f

and domain of f–1

f–1

Figure 4.9