Understanding Engineering Mathematics

Solution to review question 3.1.6

If 1< 4 x− 3 <4 then adding 3 throughout gives

1 + 3 < 4 x< 4 + 3

or 4< 4 x< 7

so 1<x<^74

3.2.7 Inverse of a function

➤

89 109➤

Suppose we are given some functiony=f(x). Then given a value ofxwe can evaluate,
in principle, the corresponding value ofy. But often we are given the reverse problem – we
knowyand want to find the value ofx. What we have in fact is an equation forxwhich
we have to solve. We call thisinverting the functionfand we write

x=f−^1 (y)

Theinverseof a functionf is the functionf−^1 such that

f−^1 (f (x))=x

i.e.
f−^1 ‘undoes’ f

It is important to note that the inverse of a function only exists if the function is ‘one-to-
one’ that is, each value ofxgives one value ofyand each value ofygives one value of
x. Sometimes this means that we have to be careful with the domains of our functions.
For example, the functiony=x^2 is only ‘one-to-one’ if, say, we restrictxto the domain
x>0. Then the inverse function isx=
√
yorf−^1 (x)=

√

x.

Example

y=f(x)=

1

x− 2

(x= 2 )

Ify=4, findx.
We have
4 =

1

x− 2

so
4 (x− 2 )= 1 = 4 x− 8

and therefore

4 x= 9

and

x=

9

4