If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

### Course: Precalculus>Unit 1

Lesson 3: Invertible functions

# Determining if a function is invertible

Learn how to build a mapping diagram for a finite function, and how to use this diagram to determine if the function is invertible. An invertible function has a one-to-one mapping between its domain and range. Functions that map multiple domain elements to the same range element are not invertible.

## Want to join the conversation?

• Doesn't the first function not work, since it fails the vertical line test?
• The vertical line test only works when you have a graph of a function within the coordinate plane.
In this video, the "graphs" are really just mapping tables/picture. The oval on the right has the input/domain values . The oval on the left has all the output/range values. Each arrow depicts a relationship between an input value and an output value. The way to determine if you have a function from these mappings is to see of any input value has more than one arrow.
• How you determine if a function is invertible
• A function is invertible if for each output value you have only one input value.
• How do you get smart?
• Practice and Consistency
• So one-to-one functions are invertible. Many-to-one functions are not invertible. What about one-to-many functions, can that be possible? I presume many-to-one and many-to-many are NOT invertible.
And, what is the difference between range, co-domain, and image?
• What you call a one-to-many function is not a function. A relation is only a function if each input has a single, definite output or set of outputs. Many-to-many relations are not functions for the same reason.

Many-to-one functions, like y=x^2 are not typically invertible unless we restrict the domain. So if we amend that we only want our outputs to be positive, we can invert y=x^2 to get y=√x. It's just that we will only get positive numbers.

And, codomain is the set of all possible numbers our function could map to. In high school, that's almost always the set of real numbers, R.

The image is the set of numbers actually mapped to. So the image of y=sin(x) is the interval [-1,1]. The image is always a subset of the codomain.

Range is a synonym for either image or codomain. Which one it's a synonym for varies between schools and people, but I think that in American high schools, it usually refers to the image.
• Is this the horizontal line test?
• Yes. If you graph the function and it passes the horizontal line test, it's invertible.
• So are all parabolas not invertible then? Since each "y" in the range would correspond to two "x"s in the domain?
(1 vote)
• Upwards and downwards opening parabolas are invertible if you restrict the domain to either side of the vertex.
• how do u solve the function f(x)=x^3 for proving that its an one-one function (i.e injective function)
(1 vote)
• Let's assume that it is invertible. If it is invertible let's try to find the form of the inverse. So we have:
f(x)=x^3=y or
x^3=y or
x=y^(1/3)
We state the function g(y)=y^(1/3). Since the symbol of the variable does not matter we can make g(x)=x^(1/3). If f and g are truly each other's inverse then f(g(x))=x for any x that belongs to the domain of g. Truly:
f(g(x))= (x^(1/3))^3=x^((1/3)*3)=x^1=x
So f is invertible. If f invertible it is also 1-1.
• so does this mean for a function to have a inverse, first it needs to be a linear(or if it's not linear, run a horizontal line(I don't know, I don't really know what it's called) then there can only be a single x value for a single y value) function so that for a single y value their can only be a single x value as well?

e.g. linear: y=3x+5
not linear(but single x for a single y): y=2^x +3
(1 vote)
• All non-horizontal linear functions are invertible, but a function does not need to be linear in order to have an inverse. There are many non-linear functions that are also invertible, such as exponential functions.

Formally speaking, there are two conditions that must be satisfied in order for a function to have an inverse.

1) A function must be injective (one-to-one). This means that for all values x and y in the domain of f, f(x) = f(y) only when x = y. So, distinct inputs will produce distinct outputs.

2) A function must be surjective (onto). This means that the codomain of f is equal to the range of f.

Any function that satisfies both of these conditions is called bijective and will always have an inverse.
• What kind of functions would be like the not invertible ones in the video? Why would different inputs ever produce the same output?
(1 vote)
• As an example the y = x² function gives the same output (y) for x and -x. For instance (-4)² = 4² = 16.