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## Precalculus

### Course: Precalculus>Unit 1

Lesson 3: Invertible functions

# 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 f takes a to b, then the inverse, f, start superscript, minus, 1, end superscript, must take b to a.

## Do all functions have an inverse function?

Consider the finite function h defined by the following table.
x1234
h, left parenthesis, x, right parenthesis2125
We can create a mapping diagram for function h.
Now let's reverse the mapping to find the inverse, h, start superscript, minus, 1, end superscript.
Notice here that h, start superscript, minus, 1, end superscript maps the input of 2 to two different outputs: 1 and 3. This means that h, start superscript, minus, 1, end superscript is not a function.
Because the inverse of h is not a function, we say that h 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 g. Notice that the inverse is indeed a function.

1) f is a finite function that is defined by this table.
xminus, 2minus, 1space, space, space, 0space, space, space, 1space, space, space, 2
f, left parenthesis, x, right parenthesis21356
Is f an invertible function?

2) g is a finite function that is defined by this table.
x2581019
g, left parenthesis, x, right parenthesisminus, 2minus, 3minus, 216
Is g an invertible function?

### Challenge Problem

3*) Is f, left parenthesis, x, right parenthesis, equals, x, squared an invertible function?

## Invertible functions and their graphs

Consider the graph of the function y, equals, x, squared.
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 y, equals, x, squared.
Take the output 4, for example. Notice that by drawing the line y, equals, 4, you can see that there are two inputs, 2 and minus, 2, associated with the output of 4.
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 y, equals, x, squared is a non-invertible function.
In contrast, consider the function y, equals, x, cubed.
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 y, equals, x, cubed is invertible.
The reasoning above describes what is called the horizontal line test: In general, a function f is invertible if it passes the horizontal line test.