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Intro to invertible functions
Not all functions have inverses. Those who do are called "invertible." Learn how we can tell whether a function is invertible or not.
Inverse functions, in the most general sense, are functions that "reverse" each other. For example, if takes to , then the inverse, , must take to .
Do all functions have an inverse function?
Consider the finite function defined by the following table.
We can create a mapping diagram for function .
Now let's reverse the mapping to find the inverse, .
Notice here that maps the input of to two different outputs: and . This means that is not a function.
Because the inverse of is not a function, we say that is non-invertible.
In general, a function is invertible only if each input has a unique output. That is, each output is paired with exactly one input. That way, when the mapping is reversed, it will still be a function!
Here's an example of an invertible function . Notice that the inverse is indeed a function.
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Challenge Problem
Invertible functions and their graphs
Consider the graph of the function .
We know that a function is invertible if each input has a unique output. Or in other words, if each output is paired with exactly one input.
But this is not the case for .
Take the output , for example. Notice that by drawing the line , you can see that there are two inputs, and , associated with the output of .
In fact, if you slide the horizontal line up and down, you will see that most outputs are associated with two inputs! So the function is a non-invertible function.
In contrast, consider the function .
If we take a horizontal line and slide it up and down the graph, it only ever intersects the function in one spot!
This means that each output corresponds with exactly one input. In other words, each input has a unique output. The function is invertible.
The reasoning above describes what is called the horizontal line test: In general, a function is invertible if it passes the horizontal line test.
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