Representing, Naming, and Evaluating Functions

Examples, solutions, and videos to help Algebra I students learn how to understand that a function from one set (called the domain) to another set (called the range) assigns each element of the domain to exactly one element of the range and understand that if f is a function and x is an element of its domain, then f(x) denotes the output f of corresponding to the input x.

Students use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

New York State Common Core Math Algebra I, Module 3, Lesson 9, Lesson 10

Worksheets for Algebra I, Module 3, Lesson 9 (pdf)
Worksheets for Algebra I, Module 3, Lesson 10 (pdf)

Lesson 9
Function: A function is a correspondence between two sets, X and Y, in which each element of X is matched1to one and only one element of Y. The set X is called the domain of the function.

The notation f: X → Y is used to name the function and describes both X and Y. If x is an element in the domain X of a function f: X → Y, then x is matched to an element of Y called f(x). We say f(x) is the value in Y that denotes the output or image of f corresponding to the input x.

Range or Image of a Function: The range (or image) of a function f: X → Y is the subset of Y, denoted f(X), defined by the following property: y is an element of f(X) if and only if there is an x in X such that f(x) = y.

Equivalent Functions: Two functions, f: X → Y and g: X → Y, are said to be equivalent (and written f = g) if they have the same domain X, take values in the same set Y, and for each x in X, f(x) = g(x).

Identity: An identity is a statement that two functions are equivalent.

Example 3

Let X = {1, 2, 3, 4} and Y = {5, 6, 7, 8, 9} . f and g are defined below.

f: X → Y
f = {(1, 7), (2, 5), (3, 6), (4, 7)}

g: Y →X
g = {(6, 4), (7, 1), (8, 1), (9, 2)}

Is f a function? If yes, what is the domain and what is the range? If no, explain why f is not a function.
Is g a function? If yes, what is the domain and range? If no, explain why g is not a function. What is f(2)?
If f(x) = 7, then what might x be?

Lesson 9 Exit Ticket

1. Given as described below.
   f:{whole numbers} → {whole numbers}
   Assign each whole number to its largest place value digit.
   For example, f(4) = 4, f(14) = 4, and f(194) = 9.
   a. What is the domain and range of f?
   b. What is f(257)?
   c. What is f(0)?
   d. What is f(999)?
   e. Find a value of that makes the equation f(x) = 7 a true statement.
2. Is the correspondence described below a function? Explain your reasoning.
   M: {women} → {people}
   Assign each woman their child.

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Lesson 10 Student Outcomes

• Students understand that a function from one set (called the domain) to another set (called the range) assigns each element of the domain to exactly one element of the range and understand that 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.
  
• Students use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

Lesson 10 Problem Set Sample Solutions

1. Let f(x) = 6x - 3 and let g(x) = 0.5(4)^x. Find the value of each function for the given input.
2. Since a variable is a placeholder, we can substitute letters that stand for numbers in for x. Let f(x) = 6x - 3 and let g(x) = 0.5(4)^x and suppose a, b, c, and h are real numbers. Find the value of each function for the given input.

Lesson 10 Problem Set Sample Solutions
4. Provide a suitable domain and range to complete the definition of each function.
6. Given the function f whose domain is the set of real numbers, let f(x) = 1 if x is a rational number, and let f(x) = 0 if x is an irrational number.

Lesson 10 Exit Ticket

1. Let f(x) = 4(3)^x. Complete the table shown below.
2. Jenna knits scarves and then sells them on Etsy, an online marketplace. Let C(x) = 4x + 20, represent the cost C in dollars to produce from 1 to 6 scarves.
   a. Create a table to show the relationship between the number of scarves x and the cost C.
   b. What are the domain and range of C?
   c. What is the meaning of C(3)?
   d. What is the meaning of the solution to the equation C(x) = 40?
   
   

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