We are asked to define the relationship between a function and its inverse. Let's start by recalling that a function pairs inputs with outputs.

(x, y) \Rightarrow f (x) = y

In the case that we want to find the input given a specific output of the function

f,

we need to define its inverse. To do so, we solve the function rule for

x

and find the reverse of the original relation.

f^{- 1} (y) = x \Rightarrow (y, x)

Notice that we switched the roles of

x

and

y

when finding the inverse function and we ended with

x .

We can also find the inverse function by switching

x

and

y

first, and then solving for

y .

Let's now examine an arbitrary function.

f (x) = 3 x - 5

To find its inverse, we will first replace

f (x)

with

y .

Then, we switch the

x

and

y

values.

Original Function y = 3 x - 5 \Rightarrow Inverse Function x = 3 y - 5

Now, we can solve it for

y .

x = 3 y - 5

AddEqn

$LHS + 5 = RHS + 5$

x + 5 = 3 y

DivEqn

$LHS / 3 = RHS / 3$

\frac{x + 5}{3} = y

RearrangeEqn

Rearrange equation

y = \frac{x + 5}{3}

Here, we will replace

y

with

f^{- 1} (x)

to have our inverse function in standard form.

y = \frac{x + 5}{3} \Rightarrow f^{- 1} (x) = \frac{x + 5}{3}

Let's now compare the two function rules.

Original Function f (x) = 3 x - 5 Inverse Function f^{- 1} (x) = \frac{x + 5}{3}

While we

multiply

by

3

and then

subtract

5

in the original function, we

add

5

and then

divide

by

3

in the inverse function. These are inverse operations used in the reverse order — which is happening because inverse functions undo each other. Now, we will make a table of values to show some inputs and outputs of these functions.

$f (x) = 3 x - 5$			$f^{- 1} (x) = \frac{x + 5}{3}$
Input	Output	$(x, y)$	Input	Output	$(x, y)$
$0$	$3 (0) - 5$	$(0, - 5)$	$- 5$	$\frac{- 5 + 5}{3}$	$(- 5, 0)$
$1$	$3 (1) - 5$	$(1, - 2)$	$- 2$	$\frac{- 2 + 5}{3}$	$(- 2, 1)$
$2$	$3 (2) - 5$	$(2, 1)$	$1$	$\frac{1 + 5}{3}$	$(1, 2)$
$3$	$3 (3) - 5$	$(3, 4)$	$4$	$\frac{4 + 5}{3}$	$(4, 3)$
$4$	$3 (4) - 5$	$(4, 7)$	$7$	$\frac{7 + 5}{3}$	$(7, 4)$
$5$	$3 (5) - 5$	$(5, 10)$	$10$	$\frac{1 0 + 5}{3}$	$(10, 5)$

We can see that the inverse function switches the input and output values of the original function.

(a, b) \Rightarrow (b, a)

Since we switch the inputs and outputs, the domain and range are also switched. In other words, the domain of the original function became the range of the inverse function and the range of the original function became the domain of the inverse function. Finally, let's examine the graphs of the functions. We can draw them using the values in the table.

As we can see, the original function

f (x)

and the inverse function

f^{- 1} (x)

are symmetric with respect to the line of reflection

y = x .

Conclusion

Let's summarize the relationship between a function and its inverse.

Inverse functions are functions that undo each other.
An inverse function switches the input and output values of the original function.
The domain and range of the inverse and the original function are interchanged.
An inverse function and the original function is symmetric with respect to the line $y = x .$

Exercise

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