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M ā Functions, Lesson 3, Domain and Range (r. 2018)
FUNCTIONS
Domain and Range
CC Standard F-IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relation- ship it describes. For example, if the function h(n) gives the number of person-hours it takes to assem- ble n engines in a factory, then the positive integers would be an appropriate domain for the function.
NG Standard AI-F.IF.5 Determine the domain of a function from its graph and, where applicable, identify the appropriate do- main for a function in context.
LEARNING OBJECTIVES
Students will be able to:
1) Determine the domain of a function from its graph.
2) Identify appropriate sets of numbers for the domain and range of a function.
Overview of Lesson Teacher Centered Introduction Overview of Lesson
**- activate studentsā prior knowledge
- vocabulary
- learning objective(s)
- big ideas: direct instruction
- modeling**
Student Centered Activities guided practice ļ§ Teacher: anticipates, monitors, selects, sequences, and connects student work
**- developing essential skills
- Regents exam questions
- formative assessment assignment (exit slip, explain the math, or journal entry)**
VOCABULARY
continuous
counting numbers
discrete
domain
integers
natural numbers
range
rational numbers
real numbers
whole numbers
BIG IDEAS
The domain of x and the range of y.
The coordinate plane consists of two perpendicular number lines, which are commonly referred to as the
x-axis and the y-axis. Each number line represents the set of real numbers. The x-axis represents the
independent variable (inputs) and the y-axis represents the dependent variable (outputs).
The domain of a function is that part (or parts) of the x-axis number line required for the funtionās input
values. This can be an interval of all real numbers, or limited to specific subsets of real numbers, such as
positive or negative integers.
The range of a function is that part (or parts) of the y-axis number line required for the funtionās output
values. This can be an interval of all real numbers, or limited to specific subsets of real numbers, such as
positive or negative integers.
A function maps an element of the domain onto one and only one element of the range.
Choosing Appropriate Domains and Ranges
Many functions make sense only when a subset of all the Real Numbers are used as inputs. This subset of
the Real Numbers that makes sense is known as the domain of the function.
Example: If a store makes $2.00 profit on each sandwich sold, total profits might be modeled by the
function , where represents total profits and s represents the number of sandwiches sold.
The entire set of real numbers, including fractions and irrational numbers, make no sense for this
function, because the store only sells whole sandwiches. In this example, the domain of the function
should be restricted to the set of whole numbers. Likewise, the range of a function can also be limited to a well-defined subset of the Real Numbers on the y-axis.
Domains and ranges can be either continuous or discrete.
ANSWERS
1. ANS: A
2. ANS: C
3. ANS: B
4. ANS: C
5. ANS: B
REGENTS EXAM QUESTIONS (through June 2018)
F.IF.B.5: Domain and Range
- The graph of the function is shown below.
The domain of the function is
If , which statement is always true?
- If , then
- (^) If , then
Let f be a function such that is defined on the domain. The range of this function is
The range of the function defined as is
The range of the function is all real numbers
less than or equal to 3) less than or equal to
greater than or equal to 4) greater than or equal to
What is the domain of the relation shown below?
If the domain of the function is , then the range is
Officials in a town use a function, C , to analyze traffic patterns. represents the rate of traffic through an intersection where n is the number of observed vehicles in a specified time interval. What would be the most appropriate domain for the function?
The function represents the height, , in feet, of an object from the ground at t seconds after it is dropped. A realistic domain for this function is
- all real numbers
Which domain would be the most appropriate set to use for a function that predicts the number of household online-devices in terms of the number of people in the household?
integers 3) irrational numbers
whole numbers 4) rational numbers
A store sells self-serve frozen yogurt sundaes. The function represents the cost, in dollars, of a sundae weighing w ounces. An appropriate domain for the function would be
integers 3) nonnegative integers
rational numbers 4) nonnegative rational numbers
A construction company uses the function , where p is the number of people working on a project, to model the amount of money it spends to complete a project. A reasonable domain for this function would be
positive integers 3) both positive and negative integers
positive real numbers 4) both positive and negative real numbers
An online company lets you download songs for $0.99 each after your have paid a $5 membership fee. Which domain would be most appropriate to calculate the cost to download songs?
rational numbers greater than zero 3) integers less than or equal to zero
whole numbers greater than or equal to one
whole numbers less than or equal to one
The daily cost of production in a factory is calculated using , where x is the number of complete products manufactured. Which set of numbers best defines the domain of?
integers 3) positive rational numbers
positive real numbers 4) whole numbers
At an ice cream shop, the profit, , is modeled by the function , where c represents the number of ice cream cones sold. An appropriate domain for this function is
an integer 3) a rational number
an integer 4) a rational number
If , which interval describes the range of this function?
Choose answer choice a because the table of values and the graph clearly show that and , and all values of y between and are between 0 and 8. Eliminate answer choice b because infinity is clearly bigger than 8. Eliminate answer choice c because these are the domain of x, not the range of y. Eliminate answer choice d because negative infinity is clearly less than 0.
PTS: 2 NAT: F.IF.A.2 TOP: Domain and Range
- ANS: 2 Strategy: Input the function in a graphing calculator and inspect the graph and table views..
The value of y approaches zero, but never reachers zero, as the value of x decreases. The the range of is.
PTS: 2 NAT: F.IF.A.2 TOP: Domain and Range KEY: real domain, exponential
- ANS: 2 Strategy: Input the function into a graphing calculator and inspect the range of y-values.
The graph and the table of values show that all values of f(x) are greater than or equal to -9. Choice b) is the correct answer.
PTS: 2 NAT: F.IF.A.2 TOP: Domain and Range KEY: real domain, quadratic
- ANS: 1 Domain refers to the x-axis while range refers to the y-axis. This question is asking what values on the x- axis are required by this relation. Strategy: Underline all the x-values of the relation, then organize the unique values.
You could graph the entire relation if you have x-values of 0, 1, and 4.
PTS: 2 NAT: F.IF.A.2 TOP: Domain and Range KEY: limited domain
- ANS: 3 Substitute each value of the domain into the function and solve for the range for each value.
PTS: 2 NAT: F.IF.A.2 TOP: Domain and Range KEY: limited domain
- ANS: 4 Strategy: Examine each answer choice and eliminate wrong answers.
Eliminate answer choices a and b because negative numbers of cars observed do not make sense. Eliminate answer choice c because fractional numbers of cars observed do not make sense. Choose answer choice d because it is the only choice that makes sense. The number of cars observed must be either zero or some counting number.
PTS: 2 NAT: F.IF.B.5 TOP: Domain and Range
- ANS: 2 Strategy: Input the function into a graphing calculator and examine it to determine a realistic range. First, transform to for input.
The graph and table of values show that it takes 3 seconds for the object to reach the ground. Therefore, a realistic domain for this function is. represents the time when the object is dropped. represents the time when the object hits the ground. Answer choice b is correct.
PTS: 2 NAT: F.IF.B.5 TOP: Domain and Range
- ANS: 2 Strategy: Eliminate wrong answers.
Eliminate answer choice a because the set of integers contains negative numbers, which do not make sense when counting the number of appliances in a household. Choose answer choice b because the set of whole numbers is defined as {0, 1, 2, 3, ...}. This does make sense when counting the number of appliances in a household.
Reason: If x represents the number of complete products manufactured, there is no need for fractions or negative numbers. Strategy: Eliminate wrong answers: a) integers There is no need for negative numbers. b) positive real numbers There is no need for fractions and/or irrational numbers. c) positive rational numbers There is no need for fractions. d) whole numbers A complete product can be represented by a whole number.
PTS: 2 NAT: F.IF.B.5 TOP: Domain and Range
- ANS: 2 Strategy: Eliminate wrong answers.
- Eliminate an integer because all of the integers less than or equal to zero are negative numbers and you cannot sell a negative number of ice cream cones.
- Select an integer because these are the whole numbers and you can only sell a whole ice cream cone.
- Eliminate a rational number because you cannot sell a negative number of ice cream cones or negatives fractions of ice cream cones.
- Eliminate a rational number because you cannot sell fractional parts of ice cream cones.
PTS: 2 NAT: F.IF.B.5 TOP: Domain and Range
- ANS: 3 Strategy: Inspect the table and graph views of this function in a graphing calculator to find the range (not the domain).
The table of values and the graph both show the smalles value of f(x) is 2, which occurs when. The maximum value of f(x) is infiniy. Therefore, the range of the function is.
NOTE: is the domain of the function. Donāt confuse domain and range.
PTS: 2 NAT: F.IF.A.2 TOP: Domain and Range KEY: real domain, quadratic
