Two functions
*f*
and
*g*
are inverse functions if
*f*
o
*g*(*x*) = *x*
and
*g*
o
*f* (*x*) = *x*
for all values of
*x*
in the domain of
*f*
and
*g*
.

For instance,
*f* (*x*) = 2*x*
and
*g*(*x*) =
*x*
are inverse functions
because
*f*
o
*g*(*x*) = *f* (*g*(*x*)) = *f* (
*x*) = 2(
*x*) = *x*
and
*g*
o
*f* (*x*) = *g*(*f* (*x*)) = *g*(2*x*) = (2*x*) = *x*
. Similarly,
*f* (*x*) = *x* + 1
and
*g*(*x*) = *x* - 1
are inverse funcions because
*f*
o
*g*(*x*) = *f* (*g*(*x*)) = *f* (*x* - 1) = (*x* - 1) + 1 = *x*
and
*g*
o
*f* (*x*) = *g*(*f* (*x*)) = *g*(*x* + 1) = (*x* + 1) - 1 = *x*
.
*h*(*x*) = 3*x* - 1
and
*j*(*x*) =
are inverse functions because
*h*
o
*j*(*x*) = *h*(*j*(*x*)) = *h*() = 3() - 1 = *x* + 1 - 1 = *x*
and
*j*
o
*h*(*x*) = *j*(*h*(*x*)) = *j*(3*x* - 1) = = = *x*
.

The inverse of a function
*f* (*x*)
is denoted
*f*
^{-1}(*x*)
.

The trick to finding the inverse of a function
*f* (*x*)
is to "undo" all the
operations on
*x*
in reverse order.

The function
*f* (*x*) = 2*x* - 4
has two steps:

- Multiply by 2.
- Subtract 4.

- Add 4.
- Divide by 2.

We can verify that this is the inverse of

f^{-1}(f(x)) =f^{-1}(2x- 4) = = =x.

f(f^{-1}(x)) =f() = 2() - 4 = (x+ 4) - 4 =x.

*Example 1*: Find the inverse of
*f* (*x*) = 3(*x* - 5)
.

Original function:

- Subtract 5.
- Multiply by 3.

- Divide by 3.
- Add 5.

Check:

f^{-1}(f(x)) =f^{-1}(3(x- 5)) = + 5 = (x- 5) + 5 =x.

f(f^{-1}(x)) =f( +5) = 3(( +5) - 5) = 3() =x.

*Example 2*: Find the inverse of
*f* (*x*) =
,
*x*≥2
(we
must restrict the domain because
*f* (*x*)
is undefined for
*x* < 2
).

Original function:

- Subract 2.
- Take the square root.

- Square.
- Add 2.

Check:

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

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

When we take the inverse of a function, the domain and range switch. In example
2, the domain of
*f*
is
*x*≥2
and the range of
*f*
is
*f* (*x*)≥ 0
. Thus,
the domain of
*f*
^{-1}
is
*x*≥ 0
and the range of
*f*
^{-1}
is
*f*
^{-1}(*x*)≥2
.