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.

Main content

Finding inverse functions

Learn how to find the formula of the inverse function of a given function. For example, find the inverse of f(x)=3x+2.
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.
Or in other words, f, left parenthesis, a, right parenthesis, equals, b, \Longleftrightarrow, f, start superscript, minus, 1, end superscript, left parenthesis, b, right parenthesis, equals, a.
In this article we will learn how to find the formula of the inverse function when we have the formula of the original function.

Before we start...

In this lesson, we will find the inverse function of f, left parenthesis, x, right parenthesis, equals, 3, x, plus, 2.
Before we do that, let's first think about how we would find f, start superscript, minus, 1, end superscript, left parenthesis, 8, right parenthesis.
To find f, start superscript, minus, 1, end superscript, left parenthesis, 8, right parenthesis, we need to find the input of f that corresponds to an output of 8. This is because if f, start superscript, minus, 1, end superscript, left parenthesis, 8, right parenthesis, equals, x, then by definition of inverses, f, left parenthesis, x, right parenthesis, equals, 8.
f(x)=3x+28=3x+2Let f(x)=86=3xSubtract 2 from both sides2=xDivide both sides by 3\begin{aligned} f(x) &= 3 x+2\\\\ 8 &= 3 x+2 &&\small{\gray{\text{Let f(x)=8}}} \\\\6&=3x &&\small{\gray{\text{Subtract 2 from both sides}}}\\\\ 2&=x &&\small{\gray{\text{Divide both sides by 3}}} \end{aligned}
So f, left parenthesis, 2, right parenthesis, equals, 8 which means that f, start superscript, minus, 1, end superscript, left parenthesis, 8, right parenthesis, equals, 2

Finding inverse functions

We can generalize what we did above to find f, start superscript, minus, 1, end superscript, left parenthesis, y, right parenthesis for any y.
To find f, start superscript, minus, 1, end superscript, left parenthesis, y, right parenthesis, we can find the input of f that corresponds to an output of y. This is because if f, start superscript, minus, 1, end superscript, left parenthesis, y, right parenthesis, equals, x then by definition of inverses, f, left parenthesis, x, right parenthesis, equals, y.
f(x)=3x+2y=3x+2Let f(x)=yy2=3xSubtract 2 from both sidesy23=xDivide both sides by 3\begin{aligned} f(x) &= 3 x+2\\\\ y &= 3 x+2 &&\small{\gray{\text{Let f(x)=y}}} \\\\y-2&=3x &&\small{\gray{\text{Subtract 2 from both sides}}}\\\\ \dfrac{y-2}{3}&=x &&\small{\gray{\text{Divide both sides by 3}}} \end{aligned}
So f, start superscript, minus, 1, end superscript, left parenthesis, y, right parenthesis, equals, start fraction, y, minus, 2, divided by, 3, end fraction.
Since the choice of the variable is arbitrary, we can write this as f, start superscript, minus, 1, end superscript, left parenthesis, x, right parenthesis, equals, start fraction, x, minus, 2, divided by, 3, end fraction.

Check your understanding

1) Linear function

Find the inverse of g, left parenthesis, x, right parenthesis, equals, 2, x, minus, 5.
g, start superscript, minus, 1, end superscript, left parenthesis, x, right parenthesis, equals

2) Cubic function

Find the inverse of h, left parenthesis, x, right parenthesis, equals, x, cubed, plus, 2.
h, start superscript, minus, 1, end superscript, left parenthesis, x, right parenthesis, equals

3) Cube-root function

Find the inverse of f, left parenthesis, x, right parenthesis, equals, 4, dot, cube root of, x, end cube root.
f, start superscript, minus, 1, end superscript, left parenthesis, x, right parenthesis, equals

4) Rational functions

Find the inverse of g, left parenthesis, x, right parenthesis, equals, start fraction, x, minus, 3, divided by, x, minus, 2, end fraction.
g, start superscript, minus, 1, end superscript, left parenthesis, x, right parenthesis, equals

5) Challenge problem

Match each function with the type of its inverse.

Want to join the conversation?