There is another method we can use to find the inverse of a function.
Taking the inverse "reverses"
*x*
and
*f* (*x*)
. Thus, in the original function,
substitute "
*x*
" for "
*f* (*x*)
" and substitute "
*f*
^{-1}(*x*)
" for "
*x*
". Then,
solve for
*f*
^{-1}(*x*)
using inverse operations in the usual manner.

*Example 1*:
*f* (*x*) =
.

- Substitute:
*x*= - Solve for
*f*^{-1}(*x*) :

5 *x*= 2( *f*^{-1}(*x*)) - 15 *x*+ 1= 2( *f*^{-1}(*x*))= *f*^{-1}(*x*)*f*^{-1}(*x*)=

- Check:
*f*^{-1}(*f*(*x*)) =*f*^{-1}() = = = =*x*

*Example 2*:
*f* (*x*) =
,
*x*≠1
,
*f* (*x*) > 0
.

- Substitute:
*x*= - Solve for
*f*^{-1}(*x*) :

*x*(*f*^{-1}(*x*) - 1)^{2}= 1 ( *f*^{-1}(*x*) - 1)^{2}= *f*^{-1}(*x*) - 1= *f*^{-1}(*x*)= + 1

- Check:
*f*^{-1}(*f*(*x*)) =*f*^{-1}( = +1 = + 1 = (*x*- 1) + 1 =*x*.

Domain of

We can also find the inverse of a function by graphing. The inverse of a
function is a reflection of that function over
the line
*y* = *x*
. In other words, all points
(*x*, *y*) = (*a*, *b*)
become
(*x*, *y*) = (*b*, *a*)
. The
*x*
and
*y*
coordinates of each point switch:

Inverse of a Graph

To find the inverse of a function, reflect the function over the line
*y* = *x*
.
Or, find several points on the graph of
*y* = *f* (*x*)
, switch their
*x*
and
*y*
coordinates, and graph the resulting points. Connect these points with a line
or curve that mirrors the line or curve of the original function.

*Example*: Find the inverse
*y* = 2*x* - 1
by graphing:

Inverse of
*y* = 2*x* - 1