Precalculus Name: 1.5: Inverse Functions Period:

INVERSE OF FUNCTIONS: Inverses UNDO things.

How would you “undo” the following? List out the steps.

Walk into a room and turn on a light

Put on socks, then shoes, then tie the laces Write the steps needed to “undo” the following. Then translate those steps into an expression.

Square x

Add five to x then divide the result by 2

An inverse relation UNDOES the original relation.

Steps to “undo” something must be done in the OPPOSITE ORDER than they were originally completed.

Notation for the inverse of a function: 1( )f x means the “inverse of ( )f x ”

For example, if 5

( )2

xf x

then 1( )f x = _____________________.

PART I. Finding an Inverse Function Algebraically

Example 1: Find an equation for 1( )f x if ( ) 3f x x .

STEP 1:

STEP 2: STEP 3: STEP 4:

Example 2: Show that 3( ) 1f x x has an inverse function and find a rule for 1( )f x . State any restrictions on

the domains of and ( )f x and 1( )f x .

NEW VOCAB

Inverse One-to-One

PART II. Finding an Inverse Graphically

3( ) 1f x x 1 3( ) 1f x x

( )f x 1( )f x

If the point (a,b) is on the graph of ( )f x , what point is on the graph of 1( )f x = _______.

So… if 𝒇(𝒂) = 𝒃, then 𝒇−𝟏(𝒃) = 𝒂.

The points (a,b) and (b,a) have reflection symmetry over _____________________.

Therefore, a function and its inverse have reflection symmetry over __________________.

Based on what we now know about reflections, graph the inverses of the functions below.

1. Is the graph at the left that of a function? _________________________________

2. How can you tell? ___________________________________________________________

3. Sketch the graph of the inverse.

4. Is the inverse a function? __________________________________________________

5. How can you tell? __________________________________________________________

6. Did you have to sketch the inverse to determine if it was going to be a function? Why or why not?

If we have a picture of the relation from the beginning, we can tell whether or not the inverse is going to be a

function by using the _____________________________________________.

If any horizontal line crosses the original function more than once, then

_________________________________________________________________________________________. Use the horizontal line test to determine which of the graphs below have inverses that are also functions?

Functions whose inverses are also functions are called __________________________________.

PART III. Verifying Inverse Functions: (The Inverse Composition Rule) Since functions and inverses UNDO each other, whatever you put into the composition of a function and its inverse, you will get out. In other words:

1( ( ))f f x x AND 1( ( ))f f x x

Example 1: Show algebraically that5

( )2

xf x

and ( ) 2 5g x x are inverse functions.

Example 2: Show algebraically that 3( ) 1f x x and 3( ) 1g x x are inverse functions.