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Related Lessons: Inverse of Linear Function Inverse of Quadratic Function Inverse of Square Root Function Inverse of Rational Function
  Verifying Two Functions are Inverses Inverse of Logarithmic Function Inverse of Exponential Function Inverse of Absolute Value Function
  Inverse of Constant Function      

 

Finding the Inverse of a Linear Function | Step by Step

 

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!
diagram showing that the domain of a function will be the range of its inverse, while its range becomes the domain of its inverse as well   diagram showing how the domain and range of a function becomes the range and domain of its inverse, respectively

 

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 "x" for domain, and "y" for range.

 

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

Key steps

1) Replace f(x) by y
2) Switch the roles of "x" and "y" , in other words, interchange x and y in the equation.
3) Solve for y in terms of x
4) Replace y by f1(x) to get the inverse function
5) *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.

 


Direction: Find the inverse functions of the following linear functions, state its domain and range if necessary.

1) example 1: f(x) = 3x-5 See solution
2) example 2: f(x)=-2x+10 See solution
3) example 3: f(x)=7x/3 -1/6 See solution
4) example 4: f(x)=x+1 for x ≥2 See solution
5) example 5: f(x)=-x+1 for -3<x<3 See solution

 


Example 1: Find the inverse of the linear functionf(x)=3x-5.

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

given function f(x)=3x-5, its inverse is f^-1(x)=(x+5)/3

That was easy!


Example 2: Find the inverse of the linear function f(x)=-2x+10.

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 − 1.

given function f(x)=-2x+10, its inverse is f^-1(x)=(10-x)/2

 


Example 3: Find the inverse of the linear function  f(x)=7x/3 - 1/6.

Some students may consider this as 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).

given function f(x)=7x/3  - 1/6, its inverse is f^-1(x)= (3x/7)+(1/14)

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


Example 4: Find the inverse of the linear function f(x)=x+1 for x ≥2, and state its domain and range.

This is a "normal" linear function, however, with a restricted domain. The allowable values of x starts at x = 2 and goes 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 on how the domain and range are determined using its graph.

 

diagram showing the domain and range of the function  f(x)=x+1  where x is greater than or equal to 2

 

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

given function f(x)=x+1, its inverse f^-1(x) = x-1

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 f(x)=-x+1 for -3<x<3 , and state its domain and range.

The first step is to plot the function in xy-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.

 

diagram showing the domain and range of f(x) = -x+1 where the domain is between the values of -3 and 3

 

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

The "x" variable in the original equation has a coefficient of −1. Keep track of this as you solve for the inverse.

given function f(x)=-x+1, its inverse f^-1(x)=1-x

 

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

 


 

Practice Problems with Answers
Worksheet 1 Worksheet 2

 

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