Main content

### Course: Get ready for AP® Calculus > Unit 3

Lesson 4: 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

## Do all functions have an inverse function?

Consider the finite function $h$ defined by the following table.

We can create a mapping diagram for function $h$ .

Now let's reverse the mapping to find the inverse, ${h}^{-1}$ .

Notice here that ${h}^{-1}$ maps the input of $2$ to two different outputs: $1$ and $3$ . This means that ${h}^{-1}$ is

*a function.***not**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.

## Check your understanding

### Challenge Problem

## Invertible functions and their graphs

Consider the graph of the function $y={x}^{2}$ .

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 $y={x}^{2}$ .

Take the output $4$ , for example. Notice that by drawing the line $y=4$ , you can see that there are two inputs, $2$ and $-2$ , 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 $y={x}^{2}$ is a

**non-invertible**function.In contrast, consider the function $y={x}^{3}$ .

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 $y={x}^{3}$ 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**## Check your understanding

## Want to join the conversation?

- I don't quiet understand what it means for a function to be invertible. Based on the examples, doesn't it mean that if different inputs create a same output means it's not invertible?(28 votes)
- Yes that is exactly correct. The reason such a function would not be invertible is because its inverse would not even be a function!(59 votes)

- So does this mean that quadratic functions are always non-invertible?(28 votes)
- Exactly. Each 𝑦 value corresponds to more than one 𝑥 value (except at the vertex). Therefore, the inverse of a quadratic function is not even a function, so quadratics are noninvertible.
*However*, there it is possible to restrict the domain of the quadratic to make it invertible.

Comment if you have questions!(34 votes)

- why does this exist again?(8 votes)
- It helps with higher level math and is useful in careers.(1 vote)

- what is it if it's not a function then?

f(x)=x^2 maps to 2 values and is considered a function. I don't understand what's the deal. How is different from it's inverse?(5 votes)- If the inverse of a function is not a function, it is just called a relation.(5 votes)

- I wonder why they didn't use the term one-to-one...(7 votes)
- I didnt get what it means to check the invertible function by having the graph y=x

what does this mean?

how do we check whether it is an invertible function or not through this??(3 votes)- If you reflect the original function, let's say f(x), over the line y=x, you get f^-1(x), or the graph of the inverse function. Then, you can use the vertical line test to check if the inverse function is really a function.

To do the vertical line test, you draw an imaginary vertical line, and if this line hits more than one point on the graph, your graph is NOT A FUNCTION.

Hope I helped!(3 votes)

- What is the difference between a normal function and invertible function? Because I am unable to understand the concept clearly even now...(3 votes)
- An invertible function is one for which we can find an inverse
*function*. Recall that a function maps its input to a unique value. For example x^2 maps 3 to 9. And only to 9.

Unfortunately it also maps -3 to 9 as well. This means that if we are told that x^2 = 9 then we can't be sure whether x was 3 or -3. It is true that square root is the inverse of squared, but it is not a function.*

e^x**is**an invertible function and ln(e^x) = x for all x.(4 votes)

- What if the function takes two inputs and outputs two outputs? Could it be invertible?(4 votes)
- Possibly, it depends on the specific function. For example, f(x, y) = <x + y, x - y> is invertible, but g(x, y) = <x + y, -x - y> is not invertible.(2 votes)

- How did we deduce that inverse functions are reflections across y=x? Does it have proof beyond that course?(2 votes)
- If a point (a, b) is on the graph of a function, then (b, a) is on the graph of its inverse. You can check that y=x is the perpendicular bisector of the line between these points, so these points are a reflection across y=x.(4 votes)

- how to know if this variable is invertible?(2 votes)
- Formally speaking, a function is invertible if and only if the function is both injective and surjective.

Let f be an injective function. For any number Y in the codomain of f there exists a number X in the domain of f such that f(X) = Y. (This is equivalent to stating that the codomain and range of the function are equal.)

Let f be a surjective function. If we pick any two different numbers X,Y in the domain of f, then f(X) will not equal f(Y). (This is equivalent to stating that every input has a unique output.)(3 votes)