What you should learn

To determine whether a given relation is a function

To find the value of a function for a given element of the domain

NCTM Curriculm Standards 2, 6 - 10

 

In doing this the teacher wants to make sure that the following words are incorporated into the introductory lesson:

Functions

Vertical line test

Functional notation

 

Introduction: During certain times of the year, airlines offer lower rates to fly to selected cities. The advertisement below shows discount ticket fares. The mileage between the cities is also listed for those people who are enrolled in frequent flier programs.

Suppose we let r represent the regular fare, d the discount fair, and m the mileage. The relation whose ordered pairs are of the form (r, d) is graphed below.

The relation whose ordered pairs are of the form (m, d) is graphed below.

Notice that in the first graph above, when r = 109, there is more than one value of d, 79 and 99. However, in the second graph above, there is exactly one value of d for each value of m. Relations with this characteristic are called functions.

Exercise 1: Determine whether each relation is a function. Explain your answer.

a. {(2, 3), (3, 0), (5, 2), (-1, -2), (4, 1)}

Since each element of the domain is paired with exactly one element of the range, this relation is a function.

 

b.

c.

x	y
4 5 5 6 -1	-1 2 3 6 1

Exercise 2: Determine whether x - 4y = 12 is a function.

Method 1: Make a table of solutions

First solve for y (y = 1/4 X -3)

 

It appears that for any given value of x, there is only one value for y that will satisfy the equation. Therefore, the equation x - 4y = 12 is a function.

 

Method 2: Graph the equation

 

Since the equation is in the form Ax + By = C, the graph of the equaiton will be a line. Graph the ordered pairs from Method 1 and connect them with a line.

 

Now place your pencil at the left of the graph to represent a vertical line. Slowly move the pencil to the right across the graph.

 

For each value of x, this vertical line passes through no more than one point on the graph. Thus, the line represents a function.

 

Using a pencil to see if a graph represents a function is one way to perform the vertical line test.

Exercise 3: Use the vertical line test to determine if each relation is a function.

a.

Since this graph does not pass the vertical line test, it is not a function.

b.

 

c.

Equations that are functions can be written in a form called functional notation. For example, consider the equation y=3x-7.

In functional notation this is represented as f(x)=3x-7

 

In a function, x represents the elements of the domain and f(x) represents the elements of the range. Suppose you want to find the value in the range that corresponds to the element 4 in the domain. This is written f(4) and is read "f of 4". The value of f(4) is found by substituting 4 for x in the equation. So, f(4) = 3(4)-7 = 5.

 

 

 

Activity: The normail systolic blood pressure S is a function of the age a of the individual. That is, a person's normal blood pressure depends on how old the person is. To determine the normal systolic blood pressure of an individual, you can use the equation S=0.5a + 110, where a represents age in years.

a. Write the equation in functional notation.

b. Find S(10), S30), S(50), and S(70)

c. Graph the function. Name the independent and dependent quantities.

d. Use the graph of the function to estimate whether blood pressure increases or decreases with age. Then estimate the blood pressure of an 80-year-old person.

 

Further Application: Attempt the following two problems:

1. If f (x) = 2x - 9, find each of the following

a. f(6)

b. f(-2)

c. f(k+1)

2. If h(z) = z-4z + 9, find each value.

a. h(-3)

b. h(5c)

c. 5[h(c)]

 

Closing Activity: Check for understanding by using this as a quick review before class is over. It should take about the last five to ten minutes. I would use it for my students as their 'ticket out the door'. Click Here.

Homework: The homework to be assigned for tonight would be: 19 - 53 odd, 54, 55, 57, 58 - 67

Alternative Homework: Enriched: 20 - 52 even, 54 - 67

Extra Practice: Students book page 768 Lesson 5-5

Extra Practice Worksheet: Click Here.

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