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Algebra 1

4.2.2 Interpreting Statements Written in Function Notation

Algebra 14.2.2 Interpreting Statements Written in Function Notation

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Activity

Let’s name the functions that relate the dog’s distance from the post and the time since its owner left: function f f for Day 1, function g g for Day 2, function h h for Day 3. The input of each function is time in seconds, t t .

Day 1

A line graph showing distance from a post in feet versus time in seconds. The curve rises and dips with a peak around 50 seconds, dips, then rises again. Two points are marked on the curve at about 0 and 60 seconds.

Day 2

A line graph showing distance from post in feet versus time in seconds. The line rises, dips, peaks, and then levels off at 4 feet from 60 seconds onward. Two points are marked at roughly (0, 1.5) and (60, 4).

Day 3

Line graph showing distance from a post in feet over time in seconds. The curve starts at 1.5 feet, peaks near 5 feet, then declines to about .5 feet. Two black dots mark data points on the curve.

Use function notation to complete the table.

Day 1 Day 2 Day 3
a. distance from post 60 seconds after the owner left
b. distance from post when the owner left
c. distance from post 150 seconds after the owner left

1. Use function notation to represent the dog's distance from the post 60 seconds after the owner left.

a. Enter your answer for Day 1:

b. Enter your answer for Day 2:

c. Enter your answer for Day 3:

2. Use function notation to represent the dog’s distance from the post when the owner left.

a. Enter your answer for Day 1:

b. Enter your answer for Day 2:

c. Enter your answer for Day 3:

3. Use function notation to represent the dog’s distance from the post 150 seconds after the owner left.

b. Enter your answer for Day 2:

c. Enter your answer for Day 3:

4. Describe what each expression represents in this context:

a. f ( 15 ) f ( 15 )

b. g ( 48 ) g ( 48 )

c. h ( t ) h ( t )

5. The equation g ( 120 ) = 4 g ( 120 ) = 4 can be interpreted to mean: “On Day 2, 120 seconds after the dog owner left, the dog was 4 feet from the post.” Describe what each equation represents in this context.

a. h ( 40 ) = 4.6 h ( 40 ) = 4.6

b. f ( t ) = 5 f ( t ) = 5

c. g ( t ) = d g ( t ) = d

Video: Interpreting Function Notation

Watch the following video to learn more about function notation.

Self Check

Self Check

A function f ( y ) = N gives the number of police officers, N , in a town in year y . What does f ( 2005 ) = 300 represent?

  1. In the year 2005, there were 300 more police officers than in 2004.
  2. In the year 2005, there were 300 police officers.
  3. At year 300, there were 2005 police officers.
  4. After 2005 years, there were 300 police officers.

Additional Resources

Understanding Function Notation

Using Function Notation

Once we determine that a relationship is a function, we need to display and define the functional relationships so that we can understand and use them, and sometimes also so that we can program them into computers. There are various ways of representing functions. A standard function notation is one representation that facilitates working with functions.

To represent “height is a function of age,” we start by identifying the descriptive variables h h for height and a a for age. The letters f f , g g , and h h are often used to represent functions, just as we use x x , y y , and z z to represent numbers and A A , B B , and C C to represent sets.

  • h h is f f of a a : We name the function f f ; height is a function of age.
  • h = f ( a ) h = f ( a ) : We use parentheses to indicate the function input.
  • f ( a ) f ( a ) : We name the function f f ; the expression is read as “ f f of a a .”

Remember, we can use any letter to name the function; the notation h ( a ) h ( a ) shows us that h h depends on a a . The value a a must be put into the function h h to get a result. The parentheses indicate that age is input into the function; they do not indicate multiplication.

We can also give an algebraic expression as the input to a function. For example, f ( a + b ) f ( a + b )   means “first add a a and b b , and the result is the input for the function f f .” The operations must be performed in this order to obtain the correct result.

Function Notation

The notation y = f ( x ) y = f ( x ) defines a function named f f . This is read as “ y y is a function of x x .” The letter x x represents the input value, or independent variable. The letter y y , or f ( x ) f ( x ) , represents the output value, or dependent variable.

Example: Using Function Notation for Days in a Month

Use function notation to represent a function whose input is the name of a month and output is the number of days in that month (don’t include leap years).

The function 31 equals f of January is where 31 is the output, f is the rule, and January is the input.

For example, f f (March) = 31 = 31 , because March has 31 days. The notation d = f ( m ) d = f ( m ) reminds us that the number of days, d d (the output), is dependent on the name of the month, m m (the input).

Analysis

Note that the inputs to a function do not have to be numbers; function inputs can be names of people, labels of geometric objects, or any other element that determines some kind of output. However, most of the functions we will work with in this course will have numbers as inputs and outputs.

Try it

Try It: Understanding Function Notation

Use function notation to express the weight of a pig, W W , in pounds as a function of its age in days, d d .

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