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 agiven function and its inverseare related.
Domain and Range are just swapped!
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 thedomainand [latex]\large{\color{red}y}[/latex] for therange.
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 functionout 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 aguide to solve for the inverse function:
That was easy!
Example 2: Find the inverse of the linear function
Towards the endpart 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
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).
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 andstate its domain and range.
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.
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 sectionabout the swapping of domainand range.
Always verify the domain and range of the inverse function using the domain and range of the original. Theyare just interchanged.
Example 5: Find the inverse of the linear function belowand state its domain and range.
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.
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
You might also like these tutorials:
- 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
Tags: Advanced Algebra, Lessons
