Applications of Inverse Functions

Lesson 2.11

Example #1 – Part A In looking over his water bills for the past year, Mr.

Aviles saw that he was charged a basic monthly fee of $7 and $4 per thousand gallons of water used. Write a function, , to show the monthly cost to Mr. Aviles, depending on how many thousands of gallons are used, .

Independent variable:

Dependent variable:

In words: The total cost is 7 dollars plus 4 dollars for every thousand gallons used.

Number of gallons used (in thousands)Monthly cost (in dollars)

Example #1 – Part B In looking over his water bills for the past year, Mr.

Aviles saw that he was charged a basic monthly fee of $7 and $4 per thousand gallons of water used. Now write a function, , to show how many thousands of gallons are used, depending on the total cost, .

Independent variable:

Dependent variable:

Number of gallons used (in thousands)

Monthly cost (in dollars)

Summary of Part A and Part B

Part A Part B

Independent Variable

Dependent Variable

Number of gallons used (in thousands)

Number of gallons used (in thousands)

Monthly cost (in dollars)

Monthly cost (in dollars)

The ______________ variable and __________ variable have switched. Thus we can conclude that and are _______________________inverse functions

independent dependent

An easier way - Using Inverses

Since these functions are inverses, use one to find the other:

Monthly cost, depending on gallons used:

Gallons used, depending on monthly cost:

SADMEG

R

𝑔 (𝑥 )= 1

4𝑥−

74

Check for Understanding What was the first step we took to solving part

a? How did we identify the independent and

dependent variables? How did we know that c(x) and g(x) were

inverses? How did we find c(x)?

Check for Understanding Turn and talk with your partner:

Describe the steps to solve this kind of problem.

SummaryIn the summary box, write a summary in your notes of the steps to solve this kind of problem.

Practice Lucy bought a flower that was 3 inches tall.

Each month that she waters it, it grows half an inch. a. Write a function, that gives the current height of the flower, based on the number of months she has been watering it, .

What does represent? What does represent?

# of months watering

𝒉 (𝒙 )=𝟏𝟐𝒙+𝟑

Current height of flower

Practiceb. Write a function, , that gives the number of months Lucy has been watering the plant, based on the height, , of the plant.

What does represent? What does represent?

c. Compose and to find and .

d. What is the relationship between and

# of months watering

Current height of flower

𝒎 (𝒙 )=𝟐 𝒙−𝟔

and are inverses of each other because h(m(x)) = m(h(x)) = x.

Examples with specific variables.

Find the inverse of

Sometimes a specific letter represents a real life item. represents Circumference represents radius

Don’t change the variables. It doesn’t make sense!!

Just solve for r!!

𝒓=𝑪𝟐𝝅

=

Stop and jot: In your summary box, write a summary of the steps we took

Turn and talk – share your summary

Independent Practice

Reminders Preview HW

Why is it important to pack up quickly?