List steps to get up and out in the morning.

List what you do to get ready for bed at night.

How does the second process compare to the first?

Inverses UNDO eachother

Traditional Lesson…

(x,y) is on f

What point must be on

f -1 (“f-inverse”) ?

(y, x)

must be on

f -1

BUT WHY???

f ( f -1 (x)) = x

and

f -1( f (x)) = x

CHECK

Both should equal x

Because inverses

“undo” eachother!

To find f -1

1. SWITCH x and y

2. SOLVE

Alternate method

1. Solve for x

2. Rename y as y-1

Example: f(x) = 2 x +8

x = 2 y + 8

2. SOLVE

x = 2 y + 8

x - 8 = 2 y

(x - 8) = y2

3. f -1 notation

(x - 8) = f -1(x)2

Find the inverse of

f (x) = 3x+2 x - 1

Find the inverse of

f (x) = x2 - 3

Find the inverse of

f (x) = 2(x - 3)2 + 5

But this is just the mechanics…

What do the standards say?

Common Core Standards

Graphical Representation

1. Sketch the graph of a FUNCTION

2. DARKEN with your pencil

3. FOLD over the dotted line, y = x

4. Scribble hard on the back of your folded paper

5. Unfold and observe

Benefits

• Allows student to observe the relationship for ALL x and y, not just discrete values

• Gives the opportunity to discuss domain vs. range in a continuous context

• Can do with ANY function

• Can use to discuss restricting the range to make the inverse a function

Graphing Inverse Functions

• Blank Inverses Exploration

(for use with handout)

• Inverses exploration

(partially done for demo)