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.

Tjek din forståelse

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?
Choose 1 answer:
Choose 1 answer:

Yes. Since each input has a unique output, the function is invertible.
The mapping diagrams below verify that this is indeed the case.
Since the inverse, f, start superscript, minus, 1, end superscript, is a function, the function f is invertible!
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?
Choose 1 answer:
Choose 1 answer:

From the table we see that g, left parenthesis, 2, right parenthesis, equals, minus, 2 and g, left parenthesis, 8, right parenthesis, equals, minus, 2. This shows that minus, 2 is not a unique output, and so function g is not invertible.
We can also use mapping diagrams to help us determine whether or not g is invertible.
Notice that g, start superscript, minus, 1, end superscript is not a function, and so g is not invertible.

Challenge Problem

3*) Is f, left parenthesis, x, right parenthesis, equals, x, start superscript, 2, end superscript an invertible function?
Choose 1 answer:
Choose 1 answer:

We can create a partial table of values for the function f, left parenthesis, x, right parenthesis, equals, x, start superscript, 2, end superscript.
xminus, 2minus, 1012
f, left parenthesis, x, right parenthesis41014
From the table, we see that f, left parenthesis, minus, 2, right parenthesis, equals, 4 and f, left parenthesis, 2, right parenthesis, equals, 4. This shows that each input does not have a unique output, and so function f is not invertible.
(Note: You could make a similar argument with other outputs as well!)

Invertible functions and their graphs

Consider the graph of the function y, equals, x, start superscript, 2, end superscript.
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, start superscript, 2, end superscript.
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, start superscript, 2, end superscript is a non-invertible function.
In contrast, consider the function y, equals, x, start superscript, 3, end superscript.
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, start superscript, 3, end superscript 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.
Recall that a function and its inverse are reflections over the line y, equals, x.
Look at what happens when y, equals, x, start superscript, 2, end superscript is reflected over the line y, equals, x.
We get a relation that is not a function, since the inverse does not pass the vertical line test!
In general, if a horizontal line intersects the graph of a function in more than one place, a vertical line will intersect the graph of its reflection over y, equals, x in more than one place. This means that the inverse will not be a function.
In contrast, the reflection of y, equals, x, start superscript, 3, end superscript does pass the vertical line test, and therefore y, equals, x, start superscript, 3, end superscript is invertible.

Tjek din forståelse

4) Is g invertible?
Choose 1 answer:
Choose 1 answer:

To determine if a function is invertible, we need to decide if each input has a unique output. To do this, we can imagine passing a horizontal line through the graph of the function.
Specifically, let's imagine a horizontal line at y, equals, 2.
Notice that the line y, equals, 2 intersects the graph of y, equals, g, left parenthesis, x, right parenthesis in three places. This means that there are three inputs that have an output of 2. So each input does not have a unique output and the function is not invertible.
Or, we could say that the function y, equals, g, left parenthesis, x, right parenthesis fails the horizontal line test, and so it is not invertible.
5) Is h invertible?
Choose 1 answer:
Choose 1 answer:

To determine if a function is invertible, we need to decide if each input has a unique output. To do this, we can imagine passing a horizontal line through the graph of the function.
If we take this horizontal line and slide it up and down the graph, it only ever intersects the function in one spot! This means that each input has a unique output and so the function is invertible.
Or, we could say that the function passes the horizontal line test, and so it is invertible.