Understanding Engineering Mathematics

In fact we can do this generally for any value ofyby finding the inverse function off:

y=f(x)=

1

x− 2

Multiplying through byx−2gives

y(x− 2 )= 1 =yx− 2 y

and so
yx= 1 + 2 y

and solving forxwe obtain

x=

1 + 2 y

y

But ify=f(x)then by definitionx=f−^1 (y)
so

x=f−^1 (y)=

1 + 2 y

y

We may also write this asf−^1 (x)=

1 + 2 x

x

, withx=0 of course.

We can now check the numerical example given above

f−^1 ( 4 )=

1 + 2 × 4

4

=

9

4

as we found previously.
The inverse of a function is often as important as the function itself. For example the
inverse of theexponential functionis thelogarithm function(Chapter 4). Graphically,
the graph of the inverse function can be obtained by reflecting the graph of the function
in the straight liney=x.
Note that the−1inf−^1 (x)does not mean forming the reciprocal – i.e.f−^1 (x)=
1 /f (x).

Solution to review question 3.1.7

Ify=f(x)=

x

x− 1

, then the inverse off(x)is given byx=f−^1 (y).

So, writing

y=

x

x− 1

we havey(x− 1 )=xor(y− 1 )x=y.

So

x=

y

y− 1

=f−^1 (y)

Hence

f−^1 (x)=

x

x− 1

x= 1