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Lesson 9
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
ALGEBRA I
Lesson 9: Representing, Naming, and Evaluating Functions
Classwork
Function: A function is a correspondence between two sets, 𝑋 and π‘Œ, in which each element of 𝑋 is matched to one and
only one element of π‘Œ. The set 𝑋 is called the domain of the function.
The notation 𝑓: 𝑋 β†’ π‘Œ is used to name the function and describes both 𝑋 and π‘Œ.
If π‘₯ is an element in the domain 𝑋 of a function 𝑓: 𝑋 β†’ π‘Œ, then π‘₯ is matched to an element of π‘Œ called 𝑓(π‘₯). We say
𝑓(π‘₯) is the value in π‘Œ that denotes the output or image of 𝑓 corresponding to the input π‘₯.
The range (or image) of a function 𝑓: 𝑋 β†’ π‘Œ is the subset of π‘Œ, denoted 𝑓(𝑋), defined by the following property: 𝑦 is an
element of 𝑓(𝑋) if and only if there is an π‘₯ in 𝑋 such that 𝑓(π‘₯) = 𝑦.
Example 1: Evaluating Whether a Relation is a function using different representations
𝒇: 𝑿 β†’ 𝒀
𝒇(𝒙) = πŸπ’™ + 𝟏
Input
Output
Describe the domain and range of the relation: ___________________________________________________
Is the relation a function? Justify your answer. ___________________________________________________
Lesson 9:
Date:
Representing, Naming, and Evaluating Functions
4/29/17
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
S.48
Lesson 9
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
ALGEBRA I
Example 2: Evaluating Whether a Relation is a function using different representations
𝒇: 𝑾 β†’ 𝒀
𝒇(π’˜) = πŸ‘π’˜πŸ
Input
Output
Describe the domain and range of the relation: ___________________________________________________
Is the relation a function? Justify your answer. ____________________________________________________________
Example 3: Evaluating Whether a Relation is a function using different representations
𝒇: 𝑿 β†’ 𝒀
𝒙 = π’šπŸ
Input
Output
Describe the domain and range of the relation: ___________________________________________________
Is the relation a function? Justify your answer. ___________________________________________________________
Lesson 9:
Date:
Representing, Naming, and Evaluating Functions
4/29/17
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
S.49
Lesson 9
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
ALGEBRA I
Example 4
Let 𝑋 = {1, 2, 3, 4} and π‘Œ = {5,6,7,8,9}. 𝑓 and 𝑔 are defined below.
𝑓: 𝑋 β†’ π‘Œ
𝑔: 𝑋 β†’ π‘Œ
𝑓 = {(1,7), (2,5), (3,6), (4,7)}
𝑔 = {(1, 5), (2, 6), (1, 8), (2,9), (3,7)}
Is 𝑓 a function? Explain why or why not.
Describe the domain and range of f.
Is 𝑔 a function? Explain why or why not.
Describe the domain and range of g.
What is 𝑓(2)?
If 𝑓(π‘₯) = 7, then what might π‘₯ be?
Example 5
The squaring function is defined as follows:
Let 𝑓: 𝑋 β†’ π‘Œ be the function such that π‘₯ ↦ π‘₯ 2 , where 𝑋 is the set of all real numbers.
𝟐
πŸ‘
What are 𝒇(𝟎), 𝒇(πŸ‘), 𝒇(βˆ’πŸ), 𝒇(βˆšπŸ‘), 𝒇(βˆ’πŸ. πŸ“), 𝒇 ( ), 𝒇(𝒂), and 𝒇(πŸ‘ + 𝒂)?
What is the range of 𝒇?
Example 5
Provide a suitable domain and range to complete the definition of each function.
b. Let 𝑓(π‘₯) = 3π‘₯
a.
Let 𝑓(π‘₯) = 4π‘₯ + 2.
c.
Let 𝐺(π‘₯) = 0.3π‘₯, where 𝐺(π‘₯) is the number of gat grams in a candy bar containing π‘₯ grams of fat.
Lesson 9:
Date:
Representing, Naming, and Evaluating Functions
4/29/17
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
S.50
Lesson 9
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
ALGEBRA I
Problem Set
1.
Which of the following are examples of a function? Justify your answers.
2.
Sequences are functions. The domain is the set of all term numbers (which is usually the positive
integers), and the range is the set of terms of the sequence. For example, the sequence 1, 4, 9, 16, 25,
36,… of perfect squares is the function:
𝐿𝑒𝑑 𝑓: {π‘π‘œπ‘ π‘–π‘‘π‘–π‘£π‘’ π‘–π‘›π‘‘π‘’π‘”π‘’π‘Ÿπ‘ } β†’ {π‘π‘’π‘Ÿπ‘“π‘’π‘π‘‘ π‘ π‘žπ‘’π‘Žπ‘Ÿπ‘’π‘ }
Assign each term number to the square of that number.
a.
What is 𝑓(3)? What does it mean?
b.
What is the solution to the equation 𝑓(π‘₯) = 49? What is the meaning of this solution?
c.
According to this definition, is βˆ’3 in the domain of f? Explain why or why not.
d.
According to this definition, is 50 in the range of f? Explain why or why not
Lesson 9:
Date:
Representing, Naming, and Evaluating Functions
4/29/17
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
S.51
Lesson 9
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
ALGEBRA I
3.
4.
Write each sequence as a function.
a.
{1, 3, 9, 27, 81, 243, … }
b.
{1, 3, 5, 7, 9, … }
c.
{-3, -6, -12, -24, …}
Let 𝑋 = {0,1,2,3,4,5}. Complete the following table using the definition of 𝑓.
𝑓: 𝑋 β†’ π‘Œ
Assign each π‘₯ in 𝑋 to the expression 2π‘₯ .
What are 𝒇(𝟎), 𝒇(𝟏), 𝒇(𝟐), 𝒇(πŸ‘), 𝒇(πŸ’), and 𝒇(πŸ“)?
What is the range of 𝒇?
5.
Write the equation for 2 different functions such that 𝑓(3) = 2.
6.
Let 𝑓(π‘₯) = 6π‘₯ βˆ’ 3, and let 𝑔(π‘₯) = 0.5(4) π‘₯ . Find the value of each function for the given input.
a.
𝒇(𝟎)
j.
π’ˆ(𝟎)
b.
𝒇(βˆ’πŸπŸŽ)
k.
π’ˆ(βˆ’πŸ)
c.
𝒇(𝟐)
l.
π’ˆ(𝟐)
d.
𝒇(𝟎. 𝟎𝟏)
m. π’ˆ(βˆ’πŸ‘)
e.
𝒇(𝟏𝟏. πŸπŸ“)
n.
π’ˆ(πŸ’)
f.
𝒇(βˆ’βˆšπŸ)
o.
π’ˆ(√𝟐)
g.
𝒇( )
πŸ“
πŸ‘
p.
π’ˆ( )
Lesson 9:
Date:
𝟏
𝟐
Representing, Naming, and Evaluating Functions
4/29/17
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
S.52
Lesson 9
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
ALGEBRA I
7.
8.
What is the range of each function given below?
a.
Let 𝑓(π‘₯) = 9π‘₯ βˆ’ 1.
b.
Let 𝑔(π‘₯) = 32π‘₯ .
c.
Let 𝑓(π‘₯) = π‘₯ 2 βˆ’ 4.
d.
Let β„Ž(π‘₯) = √π‘₯ + 2.
e.
Let π‘Ž(π‘₯) = π‘₯ + 2 such that π‘₯ is a positive integer.
f.
Let 𝑔(π‘₯) = 5π‘₯ for 0 ≀ π‘₯ ≀ 4.
Provide a suitable domain and range to complete the definition of each function.
a.
Let 𝑓(π‘₯) = 2π‘₯ + 3.
b.
Let 𝑓(π‘₯) = 2π‘₯ .
c.
Let 𝐢(π‘₯) = 9π‘₯ + 130, where 𝐢(π‘₯) is the number of calories in a sandwich containing π‘₯ grams of fat.
d.
Let 𝐡(π‘₯) = 100(2) π‘₯ , where 𝐡(π‘₯) is the number of bacteria at time π‘₯ hours over the course of one day.
Lesson 9:
Date:
Representing, Naming, and Evaluating Functions
4/29/17
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
S.53
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