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

## Definition 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 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 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 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-1 ∘ f), (f ∘ f-1) 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 (f-1f)(x) = x

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

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

## Example 5 (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.

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 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 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 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.