Inverse Functions

LessonWarm-Up and Talking Points

Included

Algebra I

Warm-Up for Inverse Function Lesson

REVIEW1. A relation is a set of _______________ ___________.

2. A function is a special relation where there is ________________________________ _______________________________________________________________________.

3. Create ordered pairs for the function displayed in the mapping.

4. Inverse Operations UNDO one another. State the inverse operation.a. Additionb. Subtractionc. Multiplicationd. Divisione. Squaring

Talk about a regular function is The inverse function is

Really you are just switching your input and output, or x and y values.

Show ordered pairs of

1( )f x

( )f x

1( )f x

-6-4-106

-20346

If you watched the Kahn Academy video, he talked through solving for the opposite variable, solving for x (which is highly unusual with a linear function … we always solve for y … think of y=mx+b, etc.) Then he showed swapping out the y for the x at the end.

That’s all fine and good, but I actually find it easier to swap the variables in the first place and solve for y (like we are used to doing).

If you understand that INVERSES undo and go backwards … (instead of DR … RD; instead of xy, yx) swap them right up front!

Suppose you are given the following directions:

• From home, go north on Rt 23 for 5 miles

• Turn east (right) onto Orchard Street

• Go to the 3rd traffic light and turn north (left) onto

Avon Drive

• Tracy’s house is the 5th house on the right.

If you start from Tracy’s house, write down the directions to get home.

How did you come up with the directions to get home from Tracy’s?

Suppose you are given the following algorithm:

• Starting with a number, add 5 to it

• Divide the result by 3

• Subtract 4 from that quantity

• Double your result

The final result is 10. Working backwards knowing this result, find the original number. Show your work.

Suppose you are given the following algorithm:

• Starting with a number, add 5 to it

• Divide the result by 3

• Subtract 4 from that quantity

• Double your result

The final result is 10. Working backwards knowing this result, find the original number. Show your work.

Write a function f(x), which when given a number x (the original number) will model the operations given above.

Write a function g(x), which when given a number x (the final result), will model the backward algorithm that you came up with above.

On the Kahn Academy video, you were shown how a function and it’s inverse reflect over the line y=x..

This is a great example. The green line is .The red line is .

( )f x1( )f x

On the Kahn Academy video, you were shown how a function and it’s inverse reflect over the line y=x..

This is a CRAZY example. The green line is .The red line is

( )f x1( )f x

Inverse Functions 

A function and its inverse function can be described as the "DO" and the "UNDO" functions.  

A function takes a starting value, performs some operation on this value, and creates an output answer. 

The inverse function takes the output answer, performs some operation on it, and arrives back at the original function's starting value.

 How to create an inverse function: Example #1

1. Change f(x) to y

2. Switch x and y

3. Solve for y

4. Use inverse notation f -1(x)

The graph of an inverse function switches the x and y values completely. 

Function graphs (x, y) Inverse Function graphs (y, x) 

It’s graph is reflected over the line y = x.

Find the algebraic inverse and graph the function and inverse on the same coordinate plane. 2. 3.

Is the inverse a function? If the inverse is not a function, how can you restrict the domain of the original function so that the inverse is also a function?

Is the inverse a function? If the inverse is not a function, how can you restrict the domain of the original function so that the inverse is also a function?

Go to Kahn Academy and watch the video for Inverse Functions Example 2 https://www.khanacademy.org/math/algebra/algebra-functions/function_inverses/v/function-inverses-example-2

Find if

 

1( )f x 2( ) ( 2) 1, for 2f x x x