MAC1105: Inverse Functions
When a task involves several steps, we may often “undo” the task by undoing the steps in reverse
order. For example, in the morning we might put on socks, and then shoes. To undo this task we
must first take off the shoes, and then the socks. If we drive from home to work by driving four
miles north and then five miles east, we may get back home by driving five miles west and then
four miles south.
To undo a multistep task, we undo the steps in reverse order.
Example 1:
For each of the following multistep tasks, write down the sequence of steps that
would “undo” the task.
a. You walk out your front door, get in your car, start the engine, and drive to the store.
b. You buy a TV stand, take it home, unpack it, put it together, and place the TV on top of it.
Solutions
a. You drive home from the store, turn off the car engine, get out of the car, and walk in your
front door.
b. You take the TV off the stand, take the stand apart, put it back in the box, take it back to the
store, and return it.
We can extend this idea to mathematical functions. For every mathematical operation, there is
another operation that “undoes” it. For example, if we add five to a number, we can subtract five
from the result to obtain our original number. If we triple a number, we can divide the result by
three to obtain our original number.
Example 2:
Find a function
g
that undoes each of the following. Verify by finding formulas for
g
f
and
f
g
.
a.
f(x)
x
5
=
+
=
b.
f(x)
3x
=
c.
f(x)
3x
5
=
+
Solutions
a. Function
f
adds 5 to the input. Define a function which subtracts 5, namely
g(x)
x
5
=

=

. In
section 5.1 we saw that the operation of composition is not generally commutative. However,
using
f
and
g
as defined above:
( ) ( ) ( )
( ) ( ) ( )
(g
f )(x)
g f(x)
g x
5
x
5
5
x
(f
g)(x)
f
g(x)
f
x
5
x
5
5
x
=
=
+
=
+

=
=
=
+
=
+

=
+
=
+
=
=

=

+
=
Not only are the results of the compositions the same, but the fact that they both simplify to
x
indicates that each operation
undoes the other. That is, subtracting five undoes the addition of
five, and adding five undoes the subtraction of five.