Inverse Functions

Inverse Functions

How to solve inverse functions problems: definition, formula, graphs, relationship between the horizontal line test, examples, and their solutions.

Definition

If you change y = f(x) with respect to x, then x = f-1(y). f-1 is the inverse function.

Here's y = f(x).

If you change this into [x = ...] form,
(with respect to x)
then x = f-1(y).

This f-1 is the inverse function.

For y = f(x),
if you put x in f,
you get y.

For x = f-1(y),
if you put y in f-1,
you get x.

This is why
f-1 is the inverse function of f.

Example 1

If f(2) = 5, find the value of f-1(5).

f(2) = 5 means
if you put 2 in f(x),
you get 5.

So, if you put 5 into f-1(y),
you get 2.

So f-1(5) = 2.

Example 2

If f(x) = 2x + 1, find the value of f-1(7).

Set f-1(7) = x.
(7 is the y value.)

Then f(x) = 7.

It says f(x) = 2x + 1.

And f(x) = 7.

So 2x + 1 = 7.

Move +1 to the right side.

Then 2x = 6.

Divide both sides by 2.

Then x = 3.

x = 3

f-1(7) = x

So f-1(7) = 3.

Example 3

For the given f(x), find f-1(x). f(x) = 2x + 4

Set f(x) = y.

It says f(x) = 2x + 4.

So 2x + 4 = y.

Write this in [x = ...] form.

Move +4 to the right side.

Then 2x = y - 4.

Divide both sides by 2.

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

This function is in [x = ...] form.

So set the right side as f-1(y).

x = (1/2)y - 2
= f-1(y)

So f-1(y) = (1/2)y - 2.

It says to find f-1(x), not f-1(y).

So change the variable to y.

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

Formula: (f-1f), (ff-1)

(f-1 of f)(x) = x, (f of f-1)(x) = x

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

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

The order of f and f-1 doesn't matter.

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

Composite functions

Example 4

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

(f-1f)(x) = x

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

You don't have to use f(x) = x2 - x + 1.

Example 5

If f(x) = -3x^2 + 19, find the value of each expression. (f of f-1)(7)

(ff-1)(x) = x

So (ff-1)(7) = 7.

Graph: y = f(x) and y = f-1(x)

y = f(x) and y = f-1(x) are symmetric about the line y = x.

y = f(x) and y = f-1(x)
are symmetric about the line y = x.

Reason:

y = f(x) is x = f-1(y).

Then y = f-1(x) is the image of x = f-1(y)
under the reflection under the line y = x,
because it shows (x, y) → (y, x).

Reflection in the line y = x

Relationship between the Horizontal Line Test

The horizontal line test for y = f(x) has the same meaning as the vertical line test for y = f-1(x). So, if y = f(x) passes the horizontal line test, then y = f(x) has an inverse function.

y = f(x) and y = f-1(x)
are symmetric about the line y = x.

So the horizontal line test for y = f(x)
has the same meaning as
the vertical line test for y = f-1(x).

So, to see if y = f-1(x) is an existing function,
do the horizontal test for y = f(x).

If y = f(x) passes the horizontal line test,
then y = f(x) is one-to-one,
and y = f(x) has an inverse function: f-1(x).

Horizontal line test

Vertical line test

Example 6

Determine whether the given function has an inverse function.

Starting from the bottom,
do the horizontal line test.

Every horizontal line shows one input.

So the function passes the horizontal line test.

Then the function is one-to-one.

The given function y = f(x)
passed the horizontal line test.

So its inverse function y = f-1(x)
passed the vertical line test.

So y = f-1(x) is an existing function.

Example 7

Determine whether the given function has an inverse function.

At the origin,
the horizontal line shows one input.

So, at the origin,
the function passes the horizontal line test.

But if you make a horizontal line like this,
it shows two inputs.

So the graph fails the horizontal line test.

So this function is not one-to-one.

The given function y = f(x)
failed the horizontal line test.

So its inverse function y = f-1(x)
failed the vertical line test.

So y = f-1(x) is not an existing function.