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?

Yes that is exactly correct. The reason such a function would not be invertible is because its inverse would not even be a function!

So does this mean that quadratic functions are always non-invertible?

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!

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?

If the inverse of a function is not a function, it is just called a relation.

What if the function takes two inputs and outputs two outputs? Could it be invertible?

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.

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??

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!

What is the difference between a normal function and invertible function? Because I am unable to understand the concept clearly even now...

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.

How did we deduce that inverse functions are reflections across y=x? Does it have proof beyond that course?

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.

how to know if this variable is invertible?

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.)

Main content

Course: Get ready for AP® Calculus > Unit 3

Lesson 4: Invertible functions

Intro to invertible functions

Google Classroom

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

b

, then the inverse,

f^{- 1}

, must take

b

a

Do all functions have an inverse function?

Consider the finite function

h

defined by the following table.

$x$ ‍	$1$ ‍	$2$ ‍	$3$ ‍	$4$ ‍
$h (x)$ ‍	$2$ ‍	$1$ ‍	$2$ ‍	$5$ ‍

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 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.

Check your understanding

f

is a finite function that is defined by this table.

$x$ ‍	$- 2$ ‍	$- 1$ ‍	$0$ ‍	$1$ ‍	$2$ ‍
$f (x)$ ‍	$2$ ‍	$1$ ‍	$3$ ‍	$5$ ‍	$6$ ‍

Is $f$ ‍ an invertible function?

g

is a finite function that is defined by this table.

$x$ ‍	$2$ ‍	$5$ ‍	$8$ ‍	$10$ ‍	$19$ ‍
$g (x)$ ‍	$- 2$ ‍	$- 3$ ‍	$- 2$ ‍	$1$ ‍	$6$ ‍

Is $g$ ‍ an invertible function?

Challenge Problem

3*) Is $f (x) = x^{2}$ ‍ an invertible function?

We can create a partial table of values for the function

f (x) = x^{2}

$x$ ‍	$- 2$ ‍	$- 1$ ‍	$0$ ‍	$1$ ‍	$2$ ‍
$f (x)$ ‍	$4$ ‍	$1$ ‍	$0$ ‍	$1$ ‍	$4$ ‍

From the table, we see that

f (- 2) = 4

and

f (2) = 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 = 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.

Recall that a function and its inverse are reflections over the line

y = x

Look at what happens when

y = x^{2}

is reflected over the line

y = 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 = x

in more than one place. This means that the inverse will not be a function.

In contrast, the reflection of

y = x^{3}

does pass the vertical line test, and therefore

y = x^{3}

is invertible.

Check your understanding

4) Is $g$ ‍ invertible?

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 = 2

Notice that the line

y = 2

intersects the graph of

y = g (x)

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 = g (x)

fails the horizontal line test, and so it is not invertible.

5) Is $h$ ‍ invertible?

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.

Want to join the conversation?

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rachelt2003
Posted 8 years ago. Direct link to rachelt2003's post “I don't quiet understand ...”
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?
Button navigates to signup pageComment on rachelt2003's post “I don't quiet understand ...”
(26 votes)
Answer
- andrewp18
  Posted 8 years ago. Direct link to andrewp18's post “Yes that is exactly corre...”
  Yes that is exactly correct. The reason such a function would not be invertible is because its inverse would not even be a function!
  Comment on andrewp18's post “Yes that is exactly corre...”
  (51 votes)
Dhruv
Posted 7 years ago. Direct link to Dhruv's post “So does this mean that qu...”
So does this mean that quadratic functions are always non-invertible?
Button navigates to signup pageComment on Dhruv's post “So does this mean that qu...”
(24 votes)
Answer
- andrewp18
  Posted 7 years ago. Direct link to andrewp18's post “Exactly. Each 𝑦 value co...”
  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!
  Comment on andrewp18's post “Exactly. Each 𝑦 value co...”
  (28 votes)
Andrea Menozzi
Posted 7 years ago. Direct link to Andrea Menozzi's post “what is it if it's not a ...”
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?
Button navigates to signup pageComment on Andrea Menozzi's post “what is it if it's not a ...”
(5 votes)
Answer
- benlagueux
  Posted 7 years ago. Direct link to benlagueux's post “If the inverse of a funct...”
  If the inverse of a function is not a function, it is just called a relation.
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  (5 votes)
Siddharth Karuturi
Posted 8 months ago. Direct link to Siddharth Karuturi's post “why does this exist again...”
why does this exist again?
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(5 votes)
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- Jacob
  Posted 8 months ago. Direct link to Jacob's post “to gatekeep u”
  to gatekeep u
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  (15 votes)
Grant Iverson
Posted 8 years ago. Direct link to Grant Iverson's post “I wonder why they didn't ...”
I wonder why they didn't use the term one-to-one...
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(6 votes)
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mrnumbersvt2012
Posted a year ago. Direct link to mrnumbersvt2012's post “What if the function take...”
What if the function takes two inputs and outputs two outputs? Could it be invertible?
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(4 votes)
Answer
- KLaudano
  Posted a year ago. Direct link to KLaudano's post “Possibly, it depends on t...”
  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.
  Comment on KLaudano's post “Possibly, it depends on t...”
  (2 votes)
Samyak
Posted 7 years ago. Direct link to Samyak's post “I didnt get what it means...”
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??
Button navigates to signup pageComment on Samyak's post “I didnt get what it means...”
(2 votes)
Answer
- Paula Zhuang
  Posted 6 years ago. Direct link to Paula Zhuang's post “If you reflect the origin...”
  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!
  Button navigates to signup page
  (2 votes)
Veronica D'Cruz
Posted 7 years ago. Direct link to Veronica D'Cruz's post “What is the difference be...”
What is the difference between a normal function and invertible function? Because I am unable to understand the concept clearly even now...
Button navigates to signup pageButton navigates to signup page
(2 votes)
Answer
- Howard Bradley
  Posted 7 years ago. Direct link to Howard Bradley's post “An invertible function is...”
  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.
  Comment on Howard Bradley's post “An invertible function is...”
  (4 votes)
Mohamed Ibrahim
Posted a year ago. Direct link to Mohamed Ibrahim's post “How did we deduce that in...”
How did we deduce that inverse functions are reflections across y=x? Does it have proof beyond that course?
Button navigates to signup pageComment on Mohamed Ibrahim's post “How did we deduce that in...”
(2 votes)
Answer
- kubleeka
  Posted a year ago. Direct link to kubleeka's post “If a point (a, b) is on t...”
  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.
  Comment on kubleeka's post “If a point (a, b) is on t...”
  (4 votes)
mark
Posted 3 years ago. Direct link to mark's post “how to know if this varia...”
how to know if this variable is invertible?
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(2 votes)
Answer
- KLaudano
  Posted 3 years ago. Direct link to KLaudano's post “Formally speaking, a func...”
  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.)
  Button navigates to signup page
  (3 votes)