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1.3 What is a Function? 9/11/23

Last updated 10 months ago
34 questions
Note from the author:
OBJECTIVES & STANDARDS
Math Objectives
  • Understand and create an expression in function notation
  • Evaluate the domain and range of a function from its constraints
  • Represent real world scenarios as a function
  • Evaluate functions for a specific input value or set of values
Common Core Math Standards
  • Link to all CCSS Math
  • CCSS.PRACTICE.MP1
  • CCSS.PRACTICE.MP4
  • CCSS.HSA.CED.A.1
  • CCSS.HSA.REI.A.1
  • CCSS.HSF.IF.A.2
  • CCSS.HSF.IF.B.5
  • CCSS.HSF.BF.A.1
Personal Finance Objectives
  • Understand how to model and calculate different types of taxes
National Standards for Personal Financial Education
Earning Income
  • 6a: Calculate the amount of taxes a person is likely to pay when given information or data about the person’s sources of income and amount of spending
  • 7b: Compare sales tax rates paid on different types of goods in their state and for online purchases
  • 8b: Compare the tax rates assessed on earned income, interest income, and capital gains income
  • 11a: Evaluate the benefits and costs of gig employment, such as driving for a cab or delivery service
OBJECTIVES & STANDARDS
Math Objectives
  • Understand and create an expression in function notation
  • Evaluate the domain and range of a function from its constraints
  • Represent real world scenarios as a function
  • Evaluate functions for a specific input value or set of values
Common Core Math Standards
  • Link to all CCSS Math
  • CCSS.PRACTICE.MP1
  • CCSS.PRACTICE.MP4
  • CCSS.HSA.CED.A.1
  • CCSS.HSA.REI.A.1
  • CCSS.HSF.IF.A.2
  • CCSS.HSF.IF.B.5
  • CCSS.HSF.BF.A.1
Personal Finance Objectives
  • Understand how to model and calculate different types of taxes
National Standards for Personal Financial Education
Earning Income
  • 6a: Calculate the amount of taxes a person is likely to pay when given information or data about the person’s sources of income and amount of spending
  • 7b: Compare sales tax rates paid on different types of goods in their state and for online purchases
  • 8b: Compare the tax rates assessed on earned income, interest income, and capital gains income
  • 11a: Evaluate the benefits and costs of gig employment, such as driving for a cab or delivery service
Intro
ACTIVITY: PLAY: What's in the Box?
Follow your teacher’s directions to complete this activity.
Learn It 1
What is a Function?
A function is a relationship where each input value has only one output value. For example:
f(x) = x2 is a function because no matter what the value of x is, x2 has only one value f(x):

If f(x)2 = x, then  is NOT a function because every x value has TWO values for f(x)


Instead of using lengthy verbal descriptions like you did in the intro activity, you can use an equation to describe what a function will do. Function notation is a way to express how the function transforms the input into the output, such as: f(x) = x + 5. This means that this function named “f” operates on the variable “x” and it adds 5 to it. If you put x ‘into’ this function, f(x) tells you what comes ‘out.’ A few things to remember about function notation:
  • The input can be represented by any variable, though ‘x’ is used most often unless another variable makes more sense for a particular problem.
  • The output is represented as f(x) which reads “the function ‘f’ of ‘x’ “ or just “f of x” for short.
  • The most common letter to define a function is ‘f’ though any could be used (e.g. g(x), h(x), m(x) )
Example 1
Johnny triples his money by investing, and then pays $2 in capital gains taxes when he sells those investments. Write this using function notation where x is the amount of money Johnny starts with.
1
Represent the steps of the function:
Write the steps in the order they happen in the function using an input variable
1) Start with x
2) Multiply x by 3
3) Subtract 2 from the result
2
Write an equation for the function:
Using parentheses initially may help keep track of the order of operations
f(x) = (x * 3) - 2
f(x) = 3x - 2
Practice It
Function Notation
Functions are a perfect way to concisely describe situations and begin to calculate outcomes and make comparisons. Practice writing functions for each of the scenarios.
1

Model the function, f, which operates on variable x by adding 2, and then doubling that quantity.

1

Jeremiah makes $15 an hour working part-time at a local fast food restaurant. He pays approximately 11% in taxes on his earnings. If h is his hours worked, model his weekly after-tax wages as a function w(h) of hours worked.

1

Sarah goes to the grocery store and buys four packs of gum. Sales tax in Connecticut is 6.35%. If g is the price of a single pack of gum, write a function p(g) to model the scenario.

1

Janet rents a booth at the farmers market that costs $150 a day. She sells jars of fresh honey from her family farm for $8 each. At the end of the day, she splits all the money she brings home with her mom who takes care of the bees. If j is the number of jars of honey sold, write a function (f(j) representing how much money Janet makes each day.

Learn It 2
Evaluating Functions
Evaluating functions is the process of finding the output value of a function for a given input value. To do this, substitute the input value for the variable and then simplify the expression.
Example 2
Maritza earns $18 an hour tutoring at the community center. On weekends, she is paid a bonus $15 per shift. How much money would Maritza earn for working 6 hours on a Saturday? If x is the number of hours worked, solve the function f(x) for x = 6 hours.
1
Express using function notation:
f(x) = 15 + 18x
2
Substitute for the input variable:
f(6) = 15 + 18(6)
3
Solve:
f(6) = 15 + 18(6)
f(6) = 15 + 108
f(6) = 123
She would earn $123 working 6 hours on Saturday

Domain and Range
The domain and range describe the limits of the function. The domain is the set of all the possible input values. The range is the set of all the possible output values. You need to know the domain BEFORE you can determine the range.
Example 3
Armando works as a street sign advertiser for a local restaurant. He is paid $10 a shift plus $0.50 for every customer who comes into the restaurant during his shift up to a maximum of $100 per shift. Determine the domain and range of the function that describes Armando’s daily wages.
1
Express using function notation:
f(x) = 10 + 0.5x
2
Determine the lower and upper bounds of the domain:
If no constraints are given by the problem, the solution is the set of all real numbers, ℝ
Lower: The fewest customers that can visit is 0
Upper: There is nothing limiting the number of customers that can visit the restaurant, so there is no upper bound
3
Determine the lower and upper bounds of the range:
Often this comes from the lowest and highest values from the input, but not always!
Lower: The lowest amount Armando can make is $10 if 0 customers visit the restaurant
Upper: If enough customers came, then Armando would hit his maximum salary of $100
4
Express the domain and range as inequalities:
If only one side is bounded you can use a simple inequality
Domain:   x > 0
Range:      10 ≤ f(x) ≤ 100
Practice It
Evaluating Functions, Domain, and Range
Functions can tell us general rules for a situation, while evaluating functions tells us what specific outcomes will result from specific inputs. Practice evaluating functions and identifying domain and range for each of the scenarios below. If no limit is specified, assume that the quantity is not bounded.
Taylor makes $13 per hour at the grocery store. She always works between 10-15 hours per week to leave time for her homework and band practice. Her gross wages can be modeled as w(h) = 13h, where h is the number of hours worked per week.
1

How much will her gross wages be if she works 12 hours this week?

1

What is the domain of this function?

1

What is the range of this function?

Susan earns 0.5% interest annually  in her savings account. She can model her savings account balance after earning interest for a year as f(x) = x + 0.005x, or just f(x) = 1.005x, where x is the account balance.
1

What will be her account balance after a year if she started with $30,000 in her account?

1

What is the domain of this function?

1

What is the range of this function?

Joan has an employee discount of 15% off at StuffMart. She needs to buy 4 packages of hot dogs and 3 packages of buns for her cookout. A package of hot dogs is $2.50 and the cheapest buns are $0.50, while the most expensive “fancy” buns are $4.50. If b is the cost of a package of buns, she can model her purchase as c(b) = 0.85(3b + 10) or c(b) = (3b +10) - 0.15(3b + 10)
1

What would her total purchase price be if buns cost $1.50?

1

What is the domain of this function?

1

What is the range of this function?

The function f operates on variable x by tripling it, adding 5, and then doubling that quantity again. This can be modeled as f(x) = 2(3x + 5).
1

What would be f(22)?

1

What would be f(-22)?

1

What is the domain of this function?

1

What is the range of this function?

APPLICATION: Prices, Discounts, and Function Notation

Level 1

Charlotte has been saving up to buy a home gym so she can work out at home without having to pay for a gym membership or worry about travel when the weather is bad (she lives in Minnesota where it snows frequently). One day, she sees an ad for a 15% off sale at a fitness equipment store and decides it’s a great time to buy the home gym! She notices the fine print says that to get the 15% off the home gym, she has to pay $139.95 up front for a 6-month subscription to a workout channel, which holds no value to her.
1

Using function notation, write an expression for the final before-tax cost, f(x), of the purchase based on the initial price, x, of the home gym equipment with this deal.

1

She was considering purchasing the AwesomeFlex3000 for $889. Using your function from question 1, what would Charlotte be Charlotte’s before-tax cost with the deal? Is it worth it for her to pay for the subscription to get 15% off? Explain your answer.

2
What is the domain_______ and range of f(x)_______ if the AwesomeFlex is the cheapest option at the store?
1

The store offers a similar deal if you purchase the $139.95 workouts but with a 20% off discount instead of 15% for purchases over $1000. The home gym she’d really like, the Shredmaster5000, costs $1199 before-tax. Write a new function, g(x) for this second deal and calculate the before-tax cost of the Shredmaster5000.

2
If Charlotte lives in Minnesota where sales tax is 6.875% applied at the end of a purchase, what is the domain_______ and range for g(x)_______ , including tax, if the ShredMaster is the cheapest system that qualifies for the second deal?

Level 2

Davis is trying to decide between two part time jobs on Saturdays. He could make $16.25 per hour working at the city’s ice arena, but getting to work and back would cost him $7.50 round trip for travel. His other option is to work as a dog walker in his neighborhood for $12.50 per hour but he can ride his bike or walk there for free. He estimates he could work up to 7 hours at the arena but up to 10 hours walking dogs. He always works at least 1 hour if he goes to the arena.
2
Write two different functions that represent Davis’s gross wages, where h is the number of hours Davis works per week, a(h) is his gross wages at the ice arena minus travel expenses, and d(h) is his gross wages from walking dogs.
a(h)=_______
d(h)=_______
4
State the domain and range for each function.
Domain of a(h):_______
Range of a(h):_______

Domain of d(h):_______
Range of d(h): _______
4
If Davis works 1 hour, what would be his gross pay at each job? What if he worked 6 hours?
a(1)=_______
d(1)=_______
a(6)=_______
d(6)=_______
2
Davis wants to know what his net pay, also called take-home pay, is going to be. Assume that 10.5% of just his paycheck, not his travel expenses, is withheld for taxes at the arena. Because dog walking is a self-employed activity, Davis estimates he needs to set aside 18% of his income for taxes. Modify your functions from question 1 to model net pay including these tax withholdings.
a(h)=_______
d(h)=_______
4
State the domain and range for each of these new functions:
Domain of a(h):_______
Range of a(h):_______

Domain of d(h):_______
Range of d(h): _______

Level 3

Part I: Rebates and Discounts José is trying to purchase a used car that he has been saving up for. When he gets to the dealership, the car salesperson tells him that there are two deals they have going on that he can choose between. One is a $350 rebate taken off the price of the vehicle, or 3.5% off the listed price.
0
Write two different expressions that represent the total cost of the vehicle with the rebate r(x) or with the percent discount p(x) as a function of the price of the vehicle, x.
Rebate r(x):_______
Percent Discount p(x): _______
0

The salesperson asks if he knows which he wants to choose before he’s even found a car. Why might José have trouble answering this question? What information would help him answer?

0

He searches the lot and finally finds a nice, modest Toyota Corolla listed at $7800. Which deal should José choose, and what will be his total cost?

Level 3
Part II: Composition of Functions
First, watch this EdPuzzle video to learn how composition of functions works.
José took a personal finance class in school where he learned how to negotiate, and he manages to talk the salesperson into giving him BOTH deals. When using both discounts, the company calculates them according to the rule: r(p(x))
0

Does this mean that the company applies the percent discount first or the coupon code first?

0

Use the rule to calculate how much José will pay for his Toyota Corolla.

0

How much would José pay for the Dodge Charger he also saw on the lot with a sticker price of $11,400?

0

How much more or less would José pay for each vehicle if the dealership instead used the rule p(r(x))?
(Toyota Corolla total compared to the Dodge Charger total).