Lesson 4 Pulling a Rabbit Out of the Hat Solidify Understanding

Jump Start

Fill in the blanks with words, numbers, or algebraic expressions that make the statement true.

If , then .

Explain why your statement is true.

Learning Focus

Understand the input-output relationship between a function and its inverse.

Find the inverse of a function.

How can we be sure that two functions are inverses?

How can we find inverse functions?

Open Up the Math: Launch, Explore, Discuss

I have a magic trick for you:

  • Pick a number, any number

  • Add

  • Multiply the result by

  • Subtract

  • Divide by

  • The answer is the number you started with!

People are often mystified by such tricks, but those of us who have studied inverse operations and inverse functions can easily figure out how they work and even create our own number tricks. Let’s get started by figuring out how inverse functions work together.

input/output diagram. x=7 to f(x)=x 8 to 7 8=15, to f^-1(x) =x-8 to 7. Inverse in words: subtract 8 from the result. Inverse in words: Subtract 8 from the result

1.

input/output diagram. x=7 to f(x)=3x to 3 times 7=21, to f^-1(x) = blank to 7. Inverse in words: blank. Inverse in words:

2.

input/output diagram. x=7 to f(x)=x^2 to 7^2=49, to f^-1(x) = blank to 7. Inverse in words: blank. Inverse in words:

Pause and Reflect

3.

input/output diagram. x=7 to f(x)=2^x to 2^7=128, to f^-1(x) = blank to 7. Inverse in words: blank. Inverse in words:

4.

input/output diagram. x=7 to f(x)=2x-5 to 2 times 7-5=9, to f^-1(x) = blank to 7. Inverse in words: blank. Inverse in words:

5.

input/output diagram. x=7 to f(x)=x 5 over 3 to 7 5 over 3=4, to f^-1(x) = blank to 7. Inverse in words: blank. Inverse in words:

6.

input/output diagram. x=7 to f(x)=(x-3)^2 to (7-3)^2, to f^-1(x) = blank to 7. Inverse in words: blank. Inverse in words:

7.

input/output diagram. x=7 to f(x)=4-square root of x to 4-square root of 7, to f^-1(x) = blank to 7. Inverse in words: blank. Inverse in words:

8.

input/output diagram. x=7 to f(x)=2^x-10 to 2^7-10=118, to f^-1(x) = blank to 7. Inverse in words: blank.