Finding the Inverse of a Linear Function

The inverse of a linear function is much easier to find as compared to other kinds of functions such as quadratic and rational. The reason is that the domain and range of a linear function naturally span all real numbers unless the domain is restricted.

Before I go over five (5) examples to illustrate the procedure, I want to show you how the domain and range of a given function and its inverse are related.

Domain and Range are just swapped!

The domain (x) of the original function is the range (y) of the inverse function, while the range (y) of the original function is the domain (x) of the inverse function.

The domain (x is greater or equal to 5 ) of the original function is the range (y is greater than or equal to 5) of the inverse function; while the range (y is less than or equal to 0) of the original function is the domain (x is less than or equal to 0) of the inverse function.

Notes to consider about the diagram:

The domain of the original function becomes the range of the inverse function.
The range of the original function becomes the domain of the inverse function.
It is customary to use the letter [latex]\large{\color{blue}x}[/latex] for the domain and [latex]\large{\color{red}y}[/latex] for the range.

The general approach on how to algebraically solve for the inverse is as follows:

Key Steps in Finding the Inverse of a Linear Function

Replace [latex]f\left( x \right)[/latex] by [latex]y[/latex].
Switch the roles of [latex]x[/latex] and [latex]y[/latex], in other words, interchange [latex]x[/latex] and [latex]y[/latex] in the equation.
Solve for [latex]y[/latex] in terms of [latex]x[/latex].
Replace [latex]y[/latex] by [latex]{f^{ – 1}}\left( x \right)[/latex] to get the inverse function.
Sometimes, it is helpful to use the domain and range of the original function to identify the correct inverse function out of two possibilities. This happens when you get a “plus or minus” case in the end.

Examples of How to Find the Inverse of a Linear Function

Example 1: Find the inverse of the linear function

This function behaves well because the domain and range are both real numbers. This ensures that its inverse must be a function too. Maybe you’re familiar with the Horizontal Line Test which guarantees that it will have an inverse whenever no horizontal line intersects or crosses the graph more than once.

Use the key steps above as a guide to solve for the inverse function:

That was easy!

Example 2: Find the inverse of the linear function

Towards the end part of the solution, I want to make the denominator positive so it looks “good”. I did it by multiplying both the numerator and denominator by [latex]-1[/latex].

Example 3: Find the inverse of the linear function

f of x is equal to 7x over 3 minus 1 over 6

Some students may consider this as a rational function because the equation contains some rational expressions. Don’t be confused by the fractions here. Yes, it has fractions however there are no variables in the denominator. This makes it just a regular linear function.

To work this out, I must get rid of the denominator. I will accomplish that by multiplying both sides of the equation by their Least Common Denominator (LCD).

y = (6x+1)/14 or (f^-1)(x) = [(3x)/7]+(1/14)

As shown above, you can write the final answers in two ways. One with a single denominator, and the other is decomposed into partial fractions.

Example 4: Find the inverse of the linear function below and state its domain and range.

f ox is equal to x plus 1 for x is greater than or equal to 2

This is a “normal” linear function, however, with a restricted domain. The allowable values of [latex]x[/latex] start at [latex]x=2[/latex] and go up to positive infinity. The range can be determined using its graph. Remember that range is the set of all y values when the acceptable values of x (domain) are substituted into the function.

Pay particular attention to how the domain and range are determined using its graph.

The domain x is greater than or equal to 2 and the range y is greater than or equal to 3, plotted on a graph

Finding the inverse of this function is really easy. But keep in mind how to correctly describe the domain and range of the inverse function. We have gone over this concept at the beginning of this section about the swapping of domain and range.

Always verify the domain and range of the inverse function using the domain and range of the original. They are just interchanged.

Example 5: Find the inverse of the linear function below and state its domain and range.

f of x is equal to negative x plus 1 for x is greater than negative 3 and less than positive 3

The first step is to plot the function in [latex]xy[/latex]-axis. Clearly label the domain and the range.

Open circle (unshaded dot) means that the number at that point is excluded. If you need to refresh on this topic, check my separate lesson about Solving Linear Inequalities.

the domain (-3<x<3) and range (-2<y<4) plotted on a graph

Secondly, find the inverse algebraically using the suggested steps. Make sure that you write the correct domain and range of the inverse function.

The [latex]x[/latex] variable in the original equation has a coefficient of [latex]-1[/latex]. Keep track of this as you solve for the inverse.

I hope that you gain some basic ideas on how to find the inverse of a linear function. I recommend that you survey the related lessons on how to find inverses of other types of functions.

You might also be interested in:

Inverse of a 2×2 Matrix

Inverse of Absolute Value Function

Inverse of Constant Function

Inverse of Exponential Function

Inverse of Logarithmic Function

Inverse of Quadratic Function

Inverse of Rational Function

Inverse of Square Root Function