Inverse Functions

Inverse Functions

How to solve inverse functions problems: how to find the inverse of a function, its property, examples, and their solutions.

Example 1

If f(x) = 2x + 4, find f^-1(x).

f-1(x) means the inverse of f(x).

To find f-1(x):

Switch x and y:
y = 2x + 4x = 2y + 4

Change the function to 'y = ':
y = (1/2)x - 2

Then f-1(x) = (1/2)x - 2.

A function and its inverse function
show the reflection in the line y = x,
because the inverse function is found by
switching x and y.

Reflection in the line y = x

Property

The composition of a function and its inverse returns the input.

If you put x into f(x), you get y.
y = f(x)

Then if you put y into f-1(x), you get x.
x = f(y), y = f-1(x)

So (f-1f)(x) = x.

Composite functions

Example 2

If f(x) = 3x - 1, find the value of each expression. 1. (f^-1(f(3))), 2. (f(f^-1(5)))

(f-1f)(x) = x

So (f-1f)(3) = 3.

(f-1f)(x) = (ff-1)(x) = x

So (ff-1)(5) = 5.

Using the Horizontal Line Test

The horizontal line test is a way to see if function shows 'one unique y, one x'. And the inverse function is found by switching x and y. So, by doing the horizontal line test, you can see if the inverse function passes the vertical line test: see if the inverse function exist.

The horizontal line test is to see if f(x) shows
'one unique y, one x'.

And f-1(x) is found by switching x and y.

So, by doing the horizontal line test,
you can see if f-1(x) passes the vertical line test:
see if f-1(x) exist.

If f(x) fails the horizontal line test,
then f-1(x) fails the vertical line test.
Then f-1(x) is not a function,
which means f-1(x) doesn't exist.

Shortly, a function has an inverse function
only if the function passes the horizontal line test.
(= is one-to-one)

Example 3

Determine whether the given function has an inverse function.

This function passes the horizontal line test.

Horizontal line test

So it has an inverse function.

Actually, you have to do the vertical line test first,
then do the horizontal line test,

because if the graph is not a 'function',
then there would be no 'inverse function',
even if it passes the horizontal line test.

In this example, it says this graph is a 'function'.
So you don't need to do the vertical line test
in this example.

Example 4

Determine whether the given function has an inverse function.

This function fails the horizontal line test.

Horizontal line test

So it doesn't have an inverse function.