Download Functions Activity 1.2 - Piscataway High School

Name____________________________ Period______ Date___________ Functions Part 2
Functions Activity 1.2: Defining Functions
You probably need to fill your car with gas more often than you would like. You commute to college
each day and to a part–time job each weekend. Your car gets good gas mileage, but the recent dramatic
fluctuation in gas prices has wreaked havoc on your budget.
1. There are two input variables that determine the cost (output) of a fill-up. What are they? Be
specific.
2. Assume you need 12.6 gallons to fill up your car. Now one of the input variables in Problem 1 will
become a constant. The value of a constant will not vary throughout the problem. The cost of a fillup is now dependent on only one variable, the price per gallon.
a. Complete the following table.
Price per Gallon, 𝑝
(dollars)
Cost of Fill-Up, 𝑐
2.00
2.50
3.00
3.50
4.00
b. Is the cost of a fill-up a function of the price per gallon? Explain.
3. a. Write a verbal statement that describes how the cost of a fill-up is determined.
b. Let 𝑝 represent the price of a gallon of gasoline pumped (input) and 𝑐 represent the cost of the
fill-up (output). Translate the verbal statement in part a into a symbolic statement (an equation)
that expresses 𝑐 in terms of 𝑝.
Defining Functions by a Symbolic Rule (Equation)
The symbolic rule (equation) 𝑐 = 12.6𝑝 is an example of a second method of defining a function. Recall
that the first method is numerical (tables and ordered pairs).
4. a. Use the given equation to determine the cost of a fill-up at a price of $3.60 per gallon.
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b. Explain the steps that you used to determine the cost in part a.
Recall from Activity 1.1 that function notation is an efficient and convenient way of representing the
output variable. The equation 𝑐 = 12.6𝑝 may be written using the function notation by replacing 𝑐 with
𝑓(𝑝) as follows.
𝑓(𝑝) = 12.6𝑝
Now, if the price per gallon is $3.60, then the cost of a fill-up can be represented by 𝑓(3.60). To
evaluate 𝑓(3.60) substitute 3.60 for 𝑝 in 𝑓(𝑝) = 12.6𝑝 as follows.
𝑓(3.60) = 12.6(3.60) = 45.36
The results can be written as 𝑓(3.60) = 45.36 or as the ordered pair (3.60, 45.36). Therefore, at a
price of $3.60 per gallon, the cost of filling your car with 12.6 gallons of gas will be $45.36.
5. a. Using function notation, write the cost if the price is $2.85 per gallon and evaluate.
Write the result as an ordered pair.
b. Use the equation for the cost-of-fill-up function to evaluate 𝑓(4.95) and write a sentence
describing its meaning. Write the result as an ordered pair.
Real Numbers
The numbers that you will be using as input and output values in this text will be real numbers. A real
number is any rational or irrational number.
A rational number is any number that can be expressed as the quotient of two integers (negative and
positive counting numbers as well as zero) such that the divisor is not zero.
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An irrational number is a real number that cannot be expressed as a quotient of two integers.
All of the numbers in Examples 1 and 2 are real numbers. A real number can be represented as a point
on the number line.
Domain and Range
6. Can any number be substituted for the input variable 𝑝 in the cost-of-fill-up function?
Describe the values of 𝑝 that make sense, and explain why they do.
Definition
The collection of all possible values of the input or independent variable is called the domain
of the function. The practical domain is the collection of replacement values of the input
variable that makes practical sense in the context of the situation.
7. a. Determine the practical domain of the cost-of-fill-up function. Refer to Problem 6.
b. Determine the domain for the general function defined by 𝑐 = 12.6𝑝, with no connection to the
context of the situation.
Definition
The collection of all possible values of the output or dependent variable is the range of the
function. The practical range corresponds to the practical domain.
8. a. What is the practical range for the cost function defined by if the practical domain is 2 to 5?
b. What is the range of this function if it has no connection to the context of the situation?
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The six pairs of values given in the table represent a function. The input or independent variable is the
year, and the output or dependent variable is the percentage. The domain of the function is
{1999, 2000, 2001, 2002, 2003, 2004} because these are all the input values. The range of the function
is {62.2, 62.9, 64.1, 64.4, 65.3}because this is the set of all of the output values. Note that although 64.4
occurs twice in the table as an output value, it is listed only once in the range.
Constructing Tables of Input/Output Values
9. Use the symbolic form of the gas cost-of-fill-up function, 𝑓(𝑝) = 12.6𝑝, to evaluate
𝑓(2), 𝑓(2.50), 𝑓(3), 𝑓(3.50), and 𝑓(4) and complete the following table. Note that the input
variable 𝑝 increases by 0.50 unit. In such a case, you say the input increases by an increment of 0.50
unit.
Price per Gallon, 𝑝
2.00
2.50
3.00
3.50
4.00
(dollars)
Cost of Fill-Up,
𝑓(𝑝)
Gross Pay Function Revisited
Recall from Activity 1.1 that if you work for an hourly wage, your gross pay is a function of the number
of hours you work.
10. a. You are working a part-time job. You work between 0 and 25 hours per week. If you earn $9.50
per hour, write an equation to determine the gross pay, g, for working h hours.
b. What is the independent variable? What is the dependent variable?
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c. Complete the following table.
Number of Hours, ℎ
Gross Pay, 𝑔, (dollars)
0
5
10
15
20
25
d. Using f for the name of the function, the output variable g can be written as 𝑔 = 𝑓(ℎ).
Rewrite the equation in part a using the function notation for gross pay.
e. What are the practical domain and the practical range of the function? Explain.
f.
Evaluate 𝑓(14) and write a sentence describing its meaning.
Summary - Defining Functions: Activity 1.2
1. Independent variable is another name for the input variable of a function.
2. Dependent variable is another name for the output variable of a function.
3. The collection of all possible replacement values for the independent or input variable is called the
domain of the function. The practical domain is the collection of replacement values of the input
variable that makes practical sense in the context of the situation.
4. The collection of all possible replacement values for the dependent or output variable is called the
range of the function. When a function describes a real situation or phenomenon, its range is
often called the practical range of the function.
5. When a function is represented by an equation, the function may also be written in function
notation. For example, given 𝑦 = 2𝑥 + 3, you can replace 𝑦 with 𝑓(𝑥) and rewrite the equation as
𝑓(𝑥) = 2𝑥 + 3.
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Name_________________
Date_________
Functions Activity 1.2: Defining Functions: Exercises
In Exercises 1 and 2,
a. Identify the independent and dependent variables.
b. Let 𝑥 represent the input variable. Use function notation to represent the output
variable.
Think what depends on what. 𝑓(𝑥) 𝑜𝑟 𝑦 𝑑𝑒𝑝𝑒𝑛𝑑𝑠 𝑜𝑛 𝑥.
1. Sales tax is a function of the price of an item. The amount of sales tax is 0.08 times the price of the
item. Use ℎ to represent the function.
a. independent (input) _____________________
b.
dependent (output) ______________________
2. The Fahrenheit measure of temperature is a function of the Celsius measure. The Fahrenheit
measure is 32 more than 9/5 times the Celsius measure. Use 𝑔 to represent the function.
a. independent (input) _____________________
b.
dependent (output) ______________________
For each function in Exercises 3–5, evaluate 𝑓(2), 𝑓(−3.2), 𝑎𝑛𝑑 𝑓(𝑎)
3. 𝑓(𝑑) = 2𝑑 – 5
4. 𝑓(𝑡) = −16𝑡 2 + 7.8𝑡 + 12
5. 𝑓(𝑥) = 4
In Exercises 6–8, construct a table of values of four ordered pairs for the given function. Fractions are
fine for output.
6. 𝑔(𝑥) = 𝑥 2 . Start the inputs at 3 and used an increment of 2.
𝑥
𝑔(𝑥)
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7. ℎ(𝑥) = 𝑥 Start the inputs at 10 and used an increment of 10.
𝑥
ℎ(𝑥)
8. 𝑓(𝑥) = 3.5𝑥 + 6. Start the inputs at 0 and used an increment of 5.
𝑥
𝑓(𝑥)
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9. a. The distance you travel while hiking is a function of how fast you hike and how long you hike at
this rate. You usually maintain a speed of 3 miles per hour while hiking. Write a verbal statement
that describes how the distance that you travel is determined.
b. Identify the input and output variables of this function.
c.
Write the verbal statement in part a using function notation for the input variable. Let 𝑡
represent the input variable. Let ℎ represent the function, and ℎ(𝑡) the output variable.
d. Which variable is the dependent variable? Explain.
e. Use the equation from part c to determine the distance traveled in 4 hours.
f.
Evaluate ℎ(7) and write a sentence describing its meaning. Write the result as an ordered pair.
g. Determine the domain and range of the general function.
h. Determine the practical domain and the practical range of the function.
i.
Generate a table of values beginning at zero with an increment of 0.5.
𝑥
𝑔(𝑥)
10. Determine the domain and range of each function.
a. {(−2,4), (0,3), (5,8), (8,11)}
b. {(−6,5), (−2,5), (0,5), (3,5)}
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11. To change a Celsius temperature to Fahrenheit, use the formula 𝐹 = 5 𝐶 + 32.
You are concerned only with temperatures from freezing to boiling.
a. What is the practical domain of the function?
b. What is the practical range of the function?
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12. The cost to own and operate a car depends on many factors including gas prices, insurance costs,
size of car, and finance charges. Using a Cost Calculator found on the Internet, you determined it
costs you approximately $0.689 per mile to own and operate a car. The total cost is a function of the
number of miles driven and can be represented by the function 𝐶 = 0.689𝑚. When you finally take
your car to the junkyard, the odometer reads 157,200 miles.
a. Identify the input variable.
b. Identify the output variable.
c. Use 𝑓 to represent the function and rewrite 𝐶 = 0.689𝑚 using function notation.
d. Evaluate 𝑓(10,000) and write a sentence describing its meaning. Write the result as an
ordered pair.
e. Create a table of values with a beginning input of 0. Use increments of 50,000 miles to
represent your costs prior to taking the car to the junkyard.
f.
What is the practical domain for this situation?
g. What is the practical range for this situation?
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