Main content
Course: Algebra (all content) > Unit 7
Lesson 19: Introduction to inverses of functions (Algebra 2 level)Intro to inverse functions
Learn what the inverse of a function is, and how to evaluate inverses of functions that are given in tables or graphs.
Inverse functions, in the most general sense, are functions that "reverse" each other.
For example, here we see that function takes to , to , and to .
The inverse of , denoted (and read as " inverse"), will reverse this mapping. Function takes to , to , and to .
Defining inverse functions
In general, if a function takes to , then the inverse function, , takes to .
From this, we have the formal definition of inverse functions:
Let's dig further into this definition by working through a couple of examples.
Example 1: Mapping diagram
Suppose function is defined by mapping diagram above. What is ?
Solution
We are given information about function and are asked a question about function . Since inverse functions reverse each other, we need to reverse our thinking.
Specifically, to find , we can find the input of whose output is . This is because if , then by definition of inverses, .
From the mapping diagram, we see that , and so .
Check your understanding
Example 2: Graph
This is the graph of function . Let's find .
Solution
To find , we can find the input of that corresponds to an output of . This is because if , then by definition of inverses, .
From the graph, we see that .
Therefore, .
Check your understanding
A graphical connection
The examples above have shown us the algebraic connection between a function and its inverse, but there is also a graphical connection!
Consider function , given in the graph and in a table of values.
We can reverse the inputs and outputs of function to find the inputs and outputs of function . So if is on the graph of , then will be on the graph of .
This gives us these graph and table of values of .
Looking at the graphs together, we see that the graph of and the graph of are reflections across the line .
This will be true in general; the graph of a function and its inverse are reflections over the line .
Check your understanding
Why study inverses?
It may seem arbitrary to be interested in inverse functions but in fact we use them all the time!
Consider that the equation can be used to convert the temperature in degrees Fahrenheit, , to a temperature in degrees Celsius, .
But suppose we wanted an equation that did the reverse – that converted a temperature in degrees Celsius to a temperature in degrees Fahrenheit. This describes the function , or the inverse function.
On a more basic level, we solve many equations in mathematics, by "isolating the variable". When we isolate the variable, we "undo" what is around it. In this way, we are using the idea of inverse functions to solve equations.
Want to join the conversation?
No posts yet.