# Graphs of Inverse FunctionsFinding Inverse Functions

The  inverse functions introduction  page showed an example of how to find an inverse function.

This page will show some more cases of finding an inverse function, along with some examples of graphing inverse functions.

To sum up the approach shown on the inverse functions intro page:

-  Change  f(x)  to  y.

-  Then solve for  x,  that is to put into the form  x =.

-  Switch over  x  and  y.

-  Change  y  to  f&nbsp-1x ).

Examples

(1.1)

Find  f&nbsp-1(x)   if   f(x) = 2x + 5

Solution

y = 2x + 5

-2x + y = 5

-2x = -y + 5            ( x -1 )

2x = y5               ( ÷ 2 )

x =

y =

=>    f
&nbsp-1(x) =

(1.2)

If    f(x) = ,   find  f&nbsp-1(5)

Solution

y               ( x (x - 3) )

y(x3) = 4

xy3y = 4

xy = 3y + 4               ( ÷ y )

x =

y =

=>    f
&nbsp-1(x) =

Now

f&nbsp-1(5)  =   =

(1.3)

For    f(x) =    ,   x ≠ -3

find   f&nbsp-1(x).

Solution

y              ( x (x + 3) )

y(x + 3) = 2x

xy + 3y = 2x

xy2x = -3y

x(y2) = -3y               ( &div (y2) )

x

y

=>    f&nbsp-1(x)  =    ,   x ≠ 2

You can sometimes see a slightly different approach demonstrated.

Where after changing  f(x)  to  y,  x &nbspand  y are switched before solving for  y, this is also an effective way to find an inverse.

## Graphs of Inverse Functions

With graphs of Inverse Functions, the main thing to remember, is that for a function  f(x), and its inverse f&nbsp-1(x), the graphs will be reflected in the line  y = x, which is shown in the image below. Consider the function  f(x) = 2x,

which has inverse function   f&nbsp-1(x) =  .

The graph of each function is a straight line, we can see what happens when we plot them.

y = 2x

 x y -1 -2 0 0 1 2

y =

 x y -2 -1 0 0 2 1

The  x  and  y  co-ordinates switch places, for a function and its inverse. ## Graphs of Inverse Functions, Examples

(1.1)

Sketch the graph of  f(x) = 3x + 2  and its inverse on the same diagram.

Solution

Inverse of  3x + 2:

y3x + 2

-3x-y + 2

3xy − 2

x      ,      f&nbsp-1(x) =

y3x + 2

 x y -1 -1 0 2 1 5

y =

 x y -1 -1 2 0 5 1 (1.2)

Sketch the graph of   f(x) = x2 − 4x       x < 2,

along with its inverse, on the same diagram.

The domain is restricted to   x < 2.  As otherwise the function would fail the Horizontal Line Test, and there wouldn't be an inverse.

Solution

yx2 − 4x

 x -2 -1 0 1 2 y 12 5 0 -3 -4

As we know that the  x  and  y  co-ordinates of the original graph and the inverse graph switch places.

We can actually go ahead with plotting and drawing both curves without needing to work out the inverse function, saving time.

The inverse curve below is in green. The curves do extend further, but the smaller section used is enough to illustrate both well enough.

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