Download Set 1 Relations Versus Functions Domain and Range

Interpreting Functions
Set 1: Relations Versus Functions/Domain and Range
Instruction
Goal: To provide opportunities for students to develop concepts and skills related to using
function notation, domain, range, relations, and functions
Common Core Standards
Functions: Interpreting Functions
Understand the concept of a function and use function notation.
F-IF.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 f is a function and x is an element of its domain, then f(x) denotes the output of f
corresponding to the input x. The graph of f is the graph of the equation y = f(x).
F-IF.2.
Use function notation, evaluate functions for inputs in their domains, and interpret
statements that use function notation in terms of a context.
Functions: Building Functions
Build a function that models a relationship between two quantities.
F-BF.1.
Write a function that describes a relationship between two quantities.‫ۻ‬
• Determine an explicit expression, a recursive process, or steps for calculation from
a context.
Student Activities Overview and Answer Key
Station 1
Students will be given eight index cards with functions and function answers on them. They will
match the functions with the appropriate function answers. Then they will evaluate functions.
Answers
2
1. f(x) = 2x with f(3) = 6; f(x) = –3t + 7 with f(3) = –2; f(x) = x2 with f(3) = 9; f ( x ) " x with
3
f(3) = 2.
2. f(x + 3) = x + 8
3. f(t – 4) = t 2 – 8t + 16 or (t – 4)(t – 4) or (t – 4)2
1
4
( s + 4)
4. f ( s + 4 ) = s + or
5
5
5
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Algebra I Station Activities for Common Core Standards
© 2011 Walch Education
Interpreting Functions
Interpreting Functions
Set 1: Relations Versus Functions/Domain and Range
Instruction
Station 2
Students will use a ruler to perform the vertical line test on graphs of relations. They will determine if
the relation is a function. They will construct a graph that is a function. Then they will determine if a
relation is a function by analyzing coordinate pairs.
Answers
1. Yes; the vertical line test holds.
2. No; the vertical line test does not hold.
3. Yes; the vertical line test holds; I used the vertical line test, which says if any vertical line passes
through a graph at more than one point, then the graph is not the graph of a function.
4. Answers will vary. Verify that the vertical line test holds.
5. Not a function because the element 3 in the domain has two assigned elements in the range.
(3, 1) and (3, 6)
6. Yes, it is a function.
Station 3
Students will be given a calculator to help them solve a real-world linear function. They will write and
solve a linear function based on two data points.
Answers
1. (100, 19), (250, 17)
1
2. slope = 75
3. Use the point (100, 19).
1
y − 19 = − ( x − 100)
75
61
61
x
x
+
+
y=−
or f ( x ) = −
75 3
75 3
4.
5.
1( 500) 61
+
= $ 13 .67
75
3
1( 60) 61
f ( 60) = −
+
= $ 19 .53
75
3
f ( 500) = −
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© 2011 Walch Education
Algebra I Station Activities for Common Core Standards
Interpreting Functions
Set 1: Relations Versus Functions/Domain and Range
Instruction
Station 4
Students will be given a number cube. They roll the number cube to populate a relation. They find
the domain and range of the relation and determine if it is a function. Then for given relations, they
determine the domain, range, and whether or not it is a function.
Answers
1. Answers will vary; verify that the domain includes the x-values.
2. Answers will vary; verify that the range includes the y-values.
3. Answers will vary; a function is a relation in which each x input has only one y output.
4. Domain: {–1, 2, 3, 4}; range: {2, 5, 10}; yes, it is a function.
5. Domain: {3, 7, 10}; range: {2, 5, 7}; not a function because there are two y-values for x = 10:
(10, 7) and (10, 5).
6. Domain: {–14, 14, 15, 17}; range: {–9, 8, 17}; yes, it is a function.
Materials List/Setup
Station 1
eight index cards with the following functions and answers written on them:
f(x) = 2x; f(x) = –3t + 7; f(x) = x2; f ( x ) "
Station 2
ruler; graph paper
Station 3
calculator
Station 4
number cube
2
x ; f(3) = 6; f(3) = 9; f(3) = –2; f(3) = 2
3
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Algebra I Station Activities for Common Core Standards
© 2011 Walch Education
Interpreting Functions
Set 1: Relations Versus Functions/Domain and Range
Instruction
Discussion Guide
To support students in reflecting on the activities and to gather some formative information about
student learning, use the following prompts to facilitate a class discussion to “debrief” the station
activities.
Prompts/Questions
1. How do you evaluate a function f(x) when given a value for x?
2. What is the vertical line test for a function?
3. What is the general formula of a linear function? How does this relate to a linear equation?
4. How do you find the domain and range of a relation?
5. How can you determine whether or not a relation is a function?
Think, Pair, Share
Have students jot down their own responses to questions, then discuss with a partner (who was not
in their station group), and then discuss as a whole class.
Suggested Appropriate Responses
1. Plug the value of x into the function to solve for f(x).
2. The vertical line test says if any vertical line passes through a graph at more than one point,
then the graph is not the graph of a function.
3. f(x) = mx + b where m and b are real numbers and m ≠ 0. This is the same as y = mx + b.
4. The domain is the x-values. The range is the y-values.
5. For every x input value, there must only be one y output value assigned to it.
Possible Misunderstandings/Mistakes
• Mixing up the domain and range
• Incorrectly thinking that in a function each y-value must have a unique x-value assigned to it
• Not keeping track of variables plugged into a function
• Using a horizontal line test instead of a vertical line test to determine if a relation is a function
265
© 2011 Walch Education
Algebra I Station Activities for Common Core Standards
NAME:
Interpreting Functions
Set 1: Relations Versus Functions/Domain and Range
Station 1
You will be given eight index cards with the following functions and answers written on them:
f(x) = 2x; f(x) = –3t + 7; f(x) = x2; f ( x ) "
2
x ; f(3) = 6; f(3) = 9; f(3) = –2; f(3) = 2
3
1. Work together to match the appropriate function with each answer. Write your matches below.
Solve. Show your work.
2. Let f(x) = x + 5. What is f(x + 3)?
3. Let f(t) = t 2. What is f(t – 4)?
4. Let f ( s ) "
1
s . What is f(s + 4)?
5
266
Algebra I Station Activities for Common Core Standards
© 2011 Walch Education
NAME:
Interpreting Functions
Set 1: Relations Versus Functions/Domain and Range
Station 2
You will be given a ruler and graph paper. As a group, use your ruler to determine whether or not
each relation below is a function. Beside each graph, write your answer and reasoning.
1.
5
y
4
3
y = 2x
2
1
–5
–4
–3
–2
–1
x
0
1
2
3
4
5
–1
–2
–3
–4
–5
2.
5
y
4
3
2
1
–5
–4
–3
–2
–1
0
x
1
2
3
4
5
–1
–2
x2 + y2 = 4
–3
–4
–5
continued
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© 2011 Walch Education
Algebra I Station Activities for Common Core Standards
NAME:
Interpreting Functions
Set 1: Relations Versus Functions/Domain and Range
3.
6
y
5
4
3
y = –3x + 6
2
1
–4
–3
–2
0
–1
x
1
2
3
4
5
6
–1
–2
–3
–4
How did you use your ruler to determine whether each relation was a function?
4. Use your ruler and graph paper to sketch a function. Use the vertical line test to verify that it
is a function.
For the relations below, determine whether or not they are functions. Explain your answer.
5. {(2, 5), (3, 1), (1, 4), (3, 6)}
6. {(1, 1), (2, 1), (3, 2)}
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Algebra I Station Activities for Common Core Standards
© 2011 Walch Education
NAME:
Interpreting Functions
Set 1: Relations Versus Functions/Domain and Range
Station 3
A function f is linear if f(x) = mx + b, where m and b are real numbers and m ≠ 0.
Use this information and the problem scenario below to answer the following questions. You may
use a calculator.
The cost of a sweatshirt is linearly related to the number of sweatshirts ordered. If you
buy 100 sweatshirts, then the cost per sweatshirt is $19. However, if you buy
250 sweatshirts, then the cost per sweatshirt is only $17.
1. You are given two points in the function. If x represents the number of sweatshirts and
y represents the cost per sweatshirt, write the two ordered pairs represented in the problem
scenario above.
2. What is the slope of the function?
3. Find a function which relates the number of sweatshirts and the cost per sweatshirt. Show your
work in the space below.
4. What would the cost per sweatshirt be for 500 sweatshirts? Explain.
5. What would the cost per sweatshirt be for 60 sweatshirts? Explain.
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© 2011 Walch Education
Algebra I Station Activities for Common Core Standards
NAME:
Interpreting Functions
Set 1: Relations Versus Functions/Domain and Range
Station 4
You will be given a number cube. As a group, roll the number cube and write the result in the first
box. Repeat this process until all the boxes contain a number.
{(
,
), (
,
), (
,
), (
,
)}
1. What is the domain of this relation?
2. What is the range of this relation?
3. Is this relation a function? Why or why not?
For problems 4–6, state the domain, range, and whether or not the relation is a function. Include
your reasoning.
4. {(2, 5), (3, 10), (–1, 2), (4, 5)}
5. {(10, 7), (3, 7), (10, 5), (7, 2)}
6. {(–14, 8), (17, 8), (14, –9), (15, 17)}
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Algebra I Station Activities for Common Core Standards
© 2011 Walch Education