Lesson 4 - Inverse
Unit 1
Chapter 1 - Introduction to Functions
LESSON 4.1 - THE INVERSE RELATION
You have been introduced to the term inverse in previous grades. For example, the inverse of addition is subtraction.
What is the inverse of
What is the inverse of
The inverse of a function is a relation which reverses or undergoes the operations performed by the function.
Consider the function f(x) = 4x + 3
The input-output diagram of f(x) is
Now, how can you get the x back from f(x)? If you undo each step in the above diagram, you will get the x back.
Common Mistake
The Inverse Relation
The Inverse Relation
Investigating Inverses
Complete the following investigation
Example #1
Solution #1
Example #2
Solution #2
LESSON 4.2 - Finding the Inverse
As we have seen earlier in this unit, functions can be presented in a variety of ways. Depending on how a function is presented, there are different methods we can use to determine the inverse. Below are each of the methods you will need to use in this course.
Equations
Given an equation, we can determine the inverse algebraically by following these steps:
Replace F(x) with y in the equation.
Replace all of the x’s with y’s and all of the y’s with x’s.
Isolate y by using the reverse operations.
After y is isolated, replace y with ƒ‾¹(x).
- Given a written definition of a function, describe and find its inverse.
- You need to reverse the operations (addition becomes subtraction, multiplication becomes division, etc.) and their order (last step becomes the first) to get the description of the inverse function.
Given the graph of a function, we can determine the inversed by reflecting the line, curve, or other function about the line y
=x.
Following these steps, we can create the graph of the inverse function.
Draw the line y=x on the graph. Recall that the inverse function is a reflection of the original function through the line y=x.
Identify several points on the function f(x). Some key points that you should always try to include are x-intercepts, y-intercepts, any points that intersect the line y=x, and any other points that you feel would be helpful to include.
uThe points that were identified in the previous step now create a series of ordered pairs, and we can use the method of ordered pairs to determine the series that represents the inverse function.
Plot the inverse series of ordered pairs on the graph and join the points with a straight line or smooth curve.
Example #1
Let f(z)={(1,5), (3,9), (4,11), (5,13)}.
a.Determine ƒ‾¹(x).
b. Determine ƒ(5) and ƒ‾¹(5).
Solution #1
Example #2
Determine the inverse of the following function.
ƒ(x)=½ x - 7
Solution #2
Example #3
Determine the inverse of the following function.
ƒ(x)=¾ x + 5
Solution #3
Example #4
Given that ƒ‾¹(1)=3, ƒ‾¹(4)=6, and ƒ(x) is a linear function, determine the equations of ƒ(x) and ƒ‾¹(x).
Solution #4
Example #5
The function ƒ(x) is given by the rule “Multiply by two and subtract seven. “Determine the inverse function.
Solution #5
Try It On Your Own
- Construct the inverse function if the original function is described by the steps:
Multiply by 3 then add 6 then divide by 2
Solution
Example #6
The function ƒ(x) is given by the rule “Multiply by two and subtract seven. “Determine the inverse function.
Solution #6
LESSON 4.3 - DOMAIN AND RANGE OF AN INVERSE
Domain and Range of an Inverse
As we know from prior lessons, the domain and range can tell us a lot about how a function looks or behaves. This is also true with inverse functions and relations.
We have already determined that there is a relationship between functions and their inverses, so we can safely assume that there must be a relationship between the domain and range of a function and the domain and range of its inverse. Let’s complete the investigation below to determine the relationship between the domain and range of a function and its inverse.
- The domain and range of the inverse
Example #1
Solution #1
Example #2
Solution #2
