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.

# Intro to inverse functions

Learn what the inverse of a function is, and how to evaluate inverses of functions that are given in tables or graphs.
Inverse functions, in the most general sense, are functions that "reverse" each other.
For example, here we see that function f takes 1 to x, 2 to z, and 3 to y.
A mapping diagram. The map is titled f. The first oval contains the values one, two, and three. The second oval contains the values x, y, and z. There is an arrow starting at one and pointing to x. There is an arrow starting at two and pointing at z. There is an arrow starting at three and pointing at y.
The inverse of f, denoted f, start superscript, minus, 1, end superscript (and read as "f inverse"), will reverse this mapping. Function f, start superscript, minus, 1, end superscript takes x to 1, y to 3, and z to 2.
A mapping diagram. The map is titled f inverse. The first oval contains the values x, y, and z. The second oval contains one, two, and three. There is an arrow starting at x and pointing to one. There is an arrow starting at y and pointing to three. There is an arrow starting z and pointing to two.
Reflection question
Which of the following is a true statement?

## Defining inverse functions

In general, if a function f takes a to b, then the inverse function, f, start superscript, minus, 1, end superscript, takes b to a.
The value a goes into function f and becomes value B which goes into f inverse and becomes value A.
From this, we have the formal definition of inverse functions:

## $f(a)=b \iff f^{-1}(b)=a$f, left parenthesis, a, right parenthesis, equals, b, \Longleftrightarrow, f, start superscript, minus, 1, end superscript, left parenthesis, b, right parenthesis, equals, a

Let's dig further into this definition by working through a couple of examples.

### Example 1: Mapping diagram

A mapping diagram. The map is titled h. The first oval contains the values zero, four, six, and nine. The second oval contains the values three, seven, nine, and twelve. There is an arrow starting at zero and pointing to seven. There is an arrow starting at four and pointing at three. There is an arrow starting at six and pointing at nine. There is an arrow starting at nine and pointing at twelve.
Suppose function h is defined by mapping diagram above. What is h, start superscript, minus, 1, end superscript, left parenthesis, 9, right parenthesis?

### Solution

We are given information about function h and are asked a question about function h, start superscript, minus, 1, end superscript. Since inverse functions reverse each other, we need to reverse our thinking.
Specifically, to find h, start superscript, minus, 1, end superscript, left parenthesis, 9, right parenthesis, we can find the input of h whose output is 9. This is because if h, start superscript, minus, 1, end superscript, left parenthesis, 9, right parenthesis, equals, x, then by definition of inverses, h, left parenthesis, x, right parenthesis, equals, 9.
From the mapping diagram, we see that h, left parenthesis, 6, right parenthesis, equals, 9, and so h, start superscript, minus, 1, end superscript, left parenthesis, 9, right parenthesis, equals, 6.

A mapping diagram. The map is titled g. The first oval contains the values negative one, zero, three, and five. The second oval contains the values two, three, four, and eight. There is an arrow starting at negative one and pointing at three. There is an arrow starting at zero and pointing at four. There is an arrow starting at three and pointing at eight. There is an arrow starting at five and pointing at two
Problem 1
g, start superscript, minus, 1, end superscript, left parenthesis, 3, right parenthesis, equals

### Example 2: Graph

This is the graph of function g. Let's find g, start superscript, minus, 1, end superscript, left parenthesis, minus, 7, right parenthesis.
A coordinate plane. The x-axis scales by zero point five, and the y-axis scales by one. The function y equals g of x is a continuous curve that starts at negative three, negative seven and increases slowly to the point negative one, negative five. Then the graph increases faster through the point zero, negative five point five and one, negative three point five. It continues to increase at a faster rate through the point two, two and the point three, ten.

### Solution

To find g, start superscript, minus, 1, end superscript, left parenthesis, minus, 7, right parenthesis, we can find the input of g that corresponds to an output of minus, 7. This is because if g, start superscript, minus, 1, end superscript, left parenthesis, minus, 7, right parenthesis, equals, x, then by definition of inverses, g, left parenthesis, x, right parenthesis, equals, minus, 7.
From the graph, we see that g, left parenthesis, minus, 3, right parenthesis, equals, minus, 7.
Therefore, g, start superscript, minus, 1, end superscript, left parenthesis, minus, 7, right parenthesis, equals, minus, 3.
A coordinate plane. The x-axis scales by zero point five, and the y-axis scales by one. The function y equals g of x is a continuous curve that starts at negative three, negative seven and increases slowly to the point negative one, negative five. Then the graph increases faster through the point zero, negative five point five and one, negative three point five. It continues to increase at a faster rate through the point two, two and the point three, ten. There is a dashed vertical line at x equals negative three and a vertical dashed line at y equals negative seven. These lines intersect at the point negative three, negative seven, which is plotted is labeled.

A coordinate plane. The x- and y- axes each scale by zero point five. The function y equals h of x is a straight that line goes through the point, negative two, four, the point zero, three, and the point two, two.
Problem 2
What is h, start superscript, minus, 1, end superscript, left parenthesis, 4, right parenthesis?

Challenge problem
Given that f, left parenthesis, x, right parenthesis, equals, 3, x, minus, 2, what is f, start superscript, minus, 1, end superscript, left parenthesis, 7, right parenthesis?

## A graphical connection

The examples above have shown us the algebraic connection between a function and its inverse, but there is also a graphical connection!
Consider function f, given in the graph and in a table of values.
A coordinate plane. The x- and y- axes each scale by one. The function y equals f of x is a nonlinear curve that goes through the following points: the point negative two, one-fourth, the point negative one, one-half, the point zero, one, the point one, two, and the point two, four.
xf, left parenthesis, x, right parenthesis
minus, 2start fraction, 1, divided by, 4, end fraction
minus, 1start fraction, 1, divided by, 2, end fraction
01
12
24
We can reverse the inputs and outputs of function f to find the inputs and outputs of function f, start superscript, minus, 1, end superscript. So if left parenthesis, a, comma, b, right parenthesis is on the graph of y, equals, f, left parenthesis, x, right parenthesis, then left parenthesis, b, comma, a, right parenthesis will be on the graph of y, equals, f, start superscript, minus, 1, end superscript, left parenthesis, x, right parenthesis.
This gives us these graph and table of values of f, start superscript, minus, 1, end superscript.
A coordinate plane. The x- and y- axes each scale by one. The function y equals f inverse of x is a nonlinear curve that goes through the following points: the point one-fourth, negative two, the point one-half, negative one, the point one, zero, the point two, one, and the point four, two.
xf, start superscript, minus, 1, end superscript, left parenthesis, x, right parenthesis
start fraction, 1, divided by, 4, end fractionminus, 2
start fraction, 1, divided by, 2, end fractionminus, 1
10
21
42
Looking at the graphs together, we see that the graph of y, equals, f, left parenthesis, x, right parenthesis and the graph of y, equals, f, start superscript, minus, 1, end superscript, left parenthesis, x, right parenthesis are reflections across the line y, equals, x.
A coordinate plane. The x- and y-axes both scale by one. There is a curved lines representing the function y equals f of x. The line is the equation y equals two to the power of x. There is another curved line representing the function y equals f inverse of x. The second line is a reflection of the first curved line over the line y equals x.
This will be true in general; the graph of a function and its inverse are reflections over the line y, equals, x.

Problem 3
This is the graph of y, equals, h, left parenthesis, x, right parenthesis.
A coordinate plane. The x- and y-axes both scale by one. There is a straight line representing the function y equals h of x. The line goes through the points zero, negative two and six, zero
Which is the best choice for the graph of y, equals, h, start superscript, minus, 1, end superscript, left parenthesis, x, right parenthesis?