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Section 3: Introduction to Functions
Topic 1: Input and Output Values ............................................................................................................................... 55
Topic 2: Representing, Naming, and Evaluating Functions ..................................................................................... 58
Topic 3: Adding and Subtracting Functions............................................................................................................... 60
Topic 4: Multiplying Functions ...................................................................................................................................... 62
Topic 5: Closure Property.............................................................................................................................................. 66
Topic 6: Real-World Combinations and Compositions of Functions ....................................................................... 68
Topic 7: Key Features of Graphs of Functions – Part 1 .............................................................................................. 71
Topic 8: Key Features of Graphs of Functions – Part 2 .............................................................................................. 74
Topic 9: Average Rate of Change Over an Interval ................................................................................................ 77
Topic 10: Transformations of Functions ....................................................................................................................... 80
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The following Mathematics Florida Standards will be covered in this section:
A-APR.1.1 - Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. A-SSE.1.1a - Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients. A-SSE.1.2 - Use the structure of an expression to identify ways to rewrite it. For example, see 𝑥𝑥 "^ – 𝑦𝑦 "^ as (𝑥𝑥 '^ ) '^ – (𝑦𝑦 '^ ) '^ thus recognizing it as a difference of squares that can be factored as (𝑥𝑥 '^ – 𝑦𝑦 '^ )(𝑥𝑥 '^ + 𝑦𝑦 '^ ). F-BF.1.1bc - Write a function that describes a relationship between two quantities. b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. c. Compose functions. For example, if 𝑇𝑇(𝑦𝑦) is the temperature in the atmosphere as a function of height, and ℎ(𝑡𝑡) is the height of a weather balloon as a function of time, then 𝑇𝑇(ℎ(𝑡𝑡)) is the temperature at the location of the weather balloon as a function of time.
F-BF.2.3 - Identify the effect on the graph of replacing 𝑓𝑓(𝑥𝑥) by 𝑓𝑓(𝑥𝑥) + 𝑘𝑘, 𝑘𝑘𝑓𝑓(𝑥𝑥), 𝑓𝑓(𝑘𝑘𝑥𝑥), and 𝑓𝑓(𝑥𝑥 + 𝑘𝑘) for specific values of 𝑘𝑘 (both positive and negative); find the value of 𝑘𝑘 given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. F-IF.1.1 - Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If 𝑓𝑓 is a function and 𝑥𝑥 is an element of its domain, then 𝑓𝑓(𝑥𝑥) denotes the output of 𝑓𝑓 corresponding to the input 𝑥𝑥. The graph of 𝑓𝑓 is the graph of the equation 𝑦𝑦 = 𝑓𝑓(𝑥𝑥). F-IF.1.2 - Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. F-IF.2.4 - For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. F-IF.2.5 - Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function ℎ(𝑛𝑛) gives the number of person-hours it takes to assemble 𝑛𝑛 engines in a factory, then the positive integers would be an appropriate domain for the function. F-IF.2.6 - Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
Let’s Practice!
- You earn $10.00 per hour babysitting. Your total earnings depend on the number of hours you spend babysitting.
a. What is the independent variable?
b. What is the dependent variable?
c. How would you represent this situation using function notation?
- The table below represents a relation.
a. Is the relation also a function? Justify your answer.
b. If the relation is not a function, what number could be changed to make it a function?
Try It!
- Mrs. Krabappel is buying composition books for her classroom. Each composition book costs $1.25.
a. What does her total cost depend upon?
b. What are the input and output?
c. Write a function to describe the situation.
d. If Mrs. Krabappel buys 24 composition books, they will cost her $30.00. Write this function using function notation.
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- Consider the following incomplete mapping diagrams.
a. Complete Diagram A so that it is a function.
b. Complete Diagram B so that it is NOT a function.
c. Is it possible to complete the mapping diagram for Diagram C so it represents a function? If so, complete the diagram to show a function. If not, justify your reasoning.
Diagram A
Diagram B
Diagram C
BEAT THE TEST!
- Isaac Messi is disorganized. To encourage Isaac to be more organized, his father promised to give him three dollars for every day that his room is clean and his schoolwork is organized.
Part A: Define the input and output for the given scenario.
Input:
Output:
Part B: Write a function to represent this situation.
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Let’s Practice!
- On the moon, the time, in seconds, it takes for an object to fall a distance, 𝑑𝑑, in feet, is given by the function 𝑓𝑓 𝑑𝑑 = 1.11 𝑑𝑑.
a. Determine 𝑓𝑓(5) and explain what it represents.
b. The South Pole-Aitken basin on the moon is 42,768 feet deep. Determine a reasonable domain for a rock dropped from the rim of the basin.
- Floyd drinks two Mountain Dew sodas in the morning. The function that represents the amount of caffeine, in milligrams, remaining in his body after drinking the sodas is given by 𝑓𝑓 𝑡𝑡 = 110 0.8855 A^ where 𝑡𝑡 is time in hours. Floyd says that in two days the caffeine is completely out of his system. Do you agree? Justify your answer.
Try It!
- Medical professionals say that 98.6℉ is the normal body temperature of an average person. Healthy individuals’ temperatures should not vary more than 0.5℉ from that temperature.
a. Write an absolute value function 𝑓𝑓(𝑡𝑡) to describe an individual’s variance from normal body temperature, where 𝑡𝑡 is the individual’s current temperature.
b. Determine 𝑓𝑓(101.5) and describe what that tells you about the individual.
c. What is a reasonable domain for a healthy individual?
BEAT THE TEST!
- The length of a shipping box is two inches longer than the width and four times the height.
Part A: Write a function 𝑉𝑉(𝑤𝑤) that models the volume of the box, where 𝑤𝑤 is the width, in inches.
Part B: Evaluate 𝑉𝑉(10). Describe what this tells you about the box.
Algebra Wall
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Section 3 – Topic 3
Adding and Subtracting Functions
Let ℎ 𝑥𝑥 = 2𝑥𝑥 >^ + 𝑥𝑥 − 5 and 𝑔𝑔 𝑥𝑥 = −3𝑥𝑥 >^ + 4𝑥𝑥 + 1.
Find ℎ 𝑥𝑥 + 𝑔𝑔(𝑥𝑥).
Find ℎ 𝑥𝑥 − 𝑔𝑔 𝑥𝑥.
BEAT THE TEST!
- Consider the functions below.
𝑓𝑓 𝑥𝑥 = 2𝑥𝑥 >^ + 3𝑥𝑥 − 5 𝑔𝑔 𝑥𝑥 = 5𝑥𝑥 >^ + 4𝑥𝑥 − 1
Which of the following is the resulting polynomial when 𝑓𝑓 𝑥𝑥 is subtracted from 𝑔𝑔(𝑥𝑥)?
A −3𝑥𝑥 >^ − 𝑥𝑥 − 4 B −3𝑥𝑥 >^ + 7𝑥𝑥 − 6 C 3𝑥𝑥 >^ + 𝑥𝑥 + 4 D 3𝑥𝑥 >^ + 7𝑥𝑥 − 6
Algebra Wall
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Section 3 – Topic 4
Multiplying Functions
Use the distributive property and modeling to perform the following function operations.
Let 𝑓𝑓 𝑥𝑥 = 3𝑥𝑥 >^ + 4𝑥𝑥 + 2 and 𝑔𝑔 𝑥𝑥 = 2𝑥𝑥 + 3.
Find 𝑓𝑓(𝑥𝑥) ∙ 𝑔𝑔(𝑥𝑥).
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Let 𝑚𝑚 𝑦𝑦 = 3 𝑦𝑦G^ − 2 𝑦𝑦>^ + 8 and 𝑝𝑝 𝑦𝑦 = 𝑦𝑦>^ − 2.
Find 𝑚𝑚(𝑦𝑦) ∙ 𝑝𝑝(𝑦𝑦).
Let’s Practice!
- Let ℎ 𝑥𝑥 = 𝑥𝑥 − 1 and 𝑔𝑔 𝑥𝑥 = 𝑥𝑥 I^ + 6𝑥𝑥 >^ − 5.
Find ℎ(𝑥𝑥) ∙ 𝑔𝑔(𝑥𝑥).
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BEAT THE TEST!
- The length of the sides of a square are 𝑠𝑠 inches long. A rectangle is six inches shorter and eight inches wider than the square.
Part A: Express both the length and the width of the rectangle as a function of a side of the square.
Part B: Write a function to represent the area of the rectangle in terms of the sides of the square.
- Felicia needs to find the area of a rectangular field in her backyard. The length is represented by the function 𝐿𝐿 𝑥𝑥 = 4𝑥𝑥 P^ − 3𝑥𝑥 >^ + 6 and the width is represented by the function 𝑊𝑊 𝑥𝑥 = 𝑥𝑥 + 1. Which of the following statements is correct about the area, 𝐴𝐴 𝑥𝑥 , of the rectangular field in Felicia’s backyard? Select all that apply.
¨ 𝐴𝐴 𝑥𝑥 = 2[𝐿𝐿 𝑥𝑥 + 𝑊𝑊 𝑥𝑥 ] ¨ The resulting expression for 𝐴𝐴(𝑥𝑥) is a fifth-degree polynomial. ¨ The resulting expression for 𝐴𝐴(𝑥𝑥) is a polynomial with a leading coefficient of 5. ¨ The resulting expression for 𝐴𝐴(𝑥𝑥) is a binomial with a constant of 6. ¨ 𝑊𝑊 𝑥𝑥 = S(T)U(T)
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Section 3 – Topic 5
Closure Property
When we add two integers, what type of number is the sum?
When we multiply two irrational numbers, what type of numbers could the resulting product be?
A set is ___________ for a specific operation if and only if the operation on two elements of the set always produces an element of the same set.
Are integers closed under addition? Justify your answer.
Are irrational numbers closed under multiplication? Justify your answer.
Are integers closed under division? Justify your answer.
Let’s apply the closure property to polynomials.
Are the following statements true or false? If false, give a counterexample.
Polynomials are closed under addition.
Polynomials are closed under subtraction.
Polynomials are closed under multiplication.
Polynomials are closed under division.
BEAT THE TEST!
- Choose from the following words and expressions to complete the statement below.
5𝑦𝑦 YZ^ + 7𝑥𝑥 >^ + 8𝑦𝑦 >
The product of 5𝑥𝑥 P^ − 3𝑥𝑥 >^ + 2 and _______________________
illustrates the closure property because the
_______________ of the product are ____________________ ,
and the product is a polynomial.
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2𝑥𝑥 G^ + (3𝑦𝑦) Y>^ − 2 (5𝑦𝑦) >^ + 4𝑥𝑥 + 3𝑦𝑦 I
integers variables whole numbers coefficients rational exponents numbers
Section 3 – Topic 6
Real-World Combinations and Compositions of
Functions
There are many times in real world situations when we must combine functions. Profit and revenue functions are a great example of this.
Let’s Practice!
- At the fall festival, the senior class sponsors hayrides to raise money for the senior trip. The ticket price is $5.00 and each hayride carries an average of 15 people. They consider raising the ticket price in order to earn more money. For each $0.50 increase in price, an average of 2 fewer seats will be sold. Let 𝑥𝑥 represent the number of $0.50 increases.
a. Write a function, 𝑇𝑇(𝑥𝑥), to represent the cost of one ticket based on the number of increases.
b. Write a function, 𝑅𝑅(𝑥𝑥), to represent the number of riders based on the number of increases.
c. Write a revenue function for the hayride that could be used to maximize revenue.
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Try It!
- The freshman class is selling t-shirts to raise money for a field trip. The cost of each custom designed t-shirt is $8. There is a $45 fee to create the design. The class plans to sell the shirts for $12.
a. Define the variable.
b. Write a cost function.
c. Write a revenue function.
d. Write a profit function.
Let’s Practice!
- Priscilla works at a cosmetics store. She receives a weekly salary of $350 and is paid a 3% commission on weekly sales over $1500.
a. Let 𝑥𝑥 represent Priscilla’s weekly sales. Write a function, 𝑓𝑓 𝑥𝑥 , to represent Priscilla’s weekly sales over $1500.
b. Let 𝑥𝑥 represent the weekly sales on which Priscilla earns commission. Write a function, 𝑔𝑔 𝑥𝑥 , to represent Priscilla’s commission.
c. Write a composite function, (𝑔𝑔 ∘ 𝑓𝑓)(𝑥𝑥) to represent the amount of money Priscilla earns on commission.
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Section 3 – Topic 7
Key Features of Graphs of Functions – Part 1
Let’s review the definition of a function.
Every input value (𝑥𝑥) corresponds to ___________ _______ output value 𝑦𝑦.
Consider the following graph.
How can a vertical line help us quickly determine if a graph represents a function?
We call this the vertical line test. Use the vertical line test to determine if the graph above represents a function.
Important facts:
Ø Graphs of lines are not always functions. Can you describe a graph of a line that is not a function?
Ø Functions are not always linear.
Sketch a graph of a function that is not linear.
3. Consider the following scenarios. Determine if each one
represents a function or not.
a. An analyst takes a survey of people about their heights (in inches) and their ages. She then relates their heights to their ages (in years).
b. A geometry student is dilating a circle and analyzes the area of the circle as it relates to the radius.
c. A teacher has a roster of 32 students and relates the students’ letter grades to the percent of points earned.
d. A boy throws a tennis ball in the air and lets it fall to the ground. The boy relates the time passed to the height of the ball.
Let’s Practice!
1. Use the vertical line test to determine if the following
graphs are functions.
Try It!
2. Which of the following graphs represent functions? Select
all that apply.
