Section 2.2

Function Notation and Linear Functions

2.2 Lecture Guide: Function Notation and Linear Functions

Objective: Use function notation.

Function Notation: f x

f xThe notation is referred to as function notation and is

is the ____________ valueread “______ of ______” or “

for an ____________ value of x.”

1.

(a)

(b)

(c)

Given 4 3f x x , evaluate each of the following:

2f

0f

7f

Objective: Use a linear equation to form a table of values and to graph a linear equation.

2. Use the function 3 4f x x the following table and graph.

Table

2

1

0

1

2

x f x

Graph

-12

4

-5 5

y

x

to complete

3. The function 3 4f x x from problem 2 is called a linear

function because its graph is a __________________ __________________. Functions in the form f x mx b are called linear functions.

4. Use the function 2 5f x x and your calculator to

complete the table below.

5. Use the function 1f x x and the Graph-Table feature

on your calculator to complete the table below. (See Calculator Perspective 2.2.2.)

2.3, 2.3, 1 by 3.1, 3.1, 1

6. The graph shown below defines a function that has an algebraic form f x mx b . Use this graph to determine the requested input and output values.

Graph

-2

8

-2 8

y

x

0f (a)(If the input value is 0, what is the output value?)

______

(b)

(c)

(d)

2f ______

0f x for x ______ (What is the input value if the output value is 0?)

for x ______ 2f x

7. The table shown below defines a function that has an algebraic form f x mx b . Use this table to determine the requested input and output values.

Table 2f (a)

(If the input value is 2, what is the output value?)

______

(b)

(c)

(d)

4f ______

2f x for x ______ (What is the input value if the output value is 2?)

for x ______ 4f x

1 2

0 0

1 2

2 4

3 6

4 8

x f x

-5

5

-1 6

8. Comparing the Graphs of an Arithmetic Sequence and a Linear Function

Graph of an Arithmetic Sequence Graph the first five terms of the sequence defined by

2 5na n .

Graph of a Linear FunctionGraph the linear function

2 5f x x by plotting thepoints with x-coordinates of 1, 2, 3, 4, and 5, and then sketch the line through these points.

na

n

-5

5

-1 6

y

x

9. Consider a car loan with payments of $200 per month and a down payment of $700.

(a) Give a function that models the total paid by the end of the xth month.

f x __________________

(b) Give the total paid by the end of the 36th month. __________________

Objective: Write a function to model an application.

10. If you make two investments totaling $3,000 and x represents the amount of one investment, write a function that represents the amount in the other investment. Then complete the table of values.

Amount of first investment Amount of second

investment

1,000

1,500

2,300

x ____________f x

11. If you have a 10-foot board that is to be cut in two pieces, and x represents the length of one of the pieces, write a function that represents the length of the other piece. Then complete the table of values.

Length of first piece Length of second piece

1

7

8

x ____________f x

12. You have 40 feet of fencing to enclose three sides of a rectangular pen, and x represents the width of the pen. Write a function that represents the length of the pen. Then complete the table of values.

Width Length

5

10

15

x ____________f x

wall

xx

13. If you have a 2 gallons of insecticide to which you are planning to add some water to dilute the mixture. Letting x represent the number of gallons of water that you add, write a function that represents the total volume in gallons of the mixture. Then complete the table of values.

Gallons of water Total volume

5

8

12

x ____________f x

14. The price of every item in a store has been marked down by 10%. Let x represent the original price of an item.

Original price New price

22

45

80

x ____________f x

(a) Write a function for the amount of discount on an item with an original price of x dollars.

(b) Write a function for the new price of an item with an original price of x dollars.

(c) Complete the following table for the new price of each item whose original price is given.

15. An airplane has a speed of x mi/h in calm skies.

(a) Write a function in terms of x for the rate of this airplane traveling in the same direction as a 20-mi/h wind.

(b) Write a function in terms of x for the rate of this airplane traveling in the opposite direction of a 20-mi/h wind.

(c) Write a function in terms of x for the distance the airplane travels in 3 hours going in the same direction as a 20-mi/h wind.

(d) Evaluate the function in part (c) using 230x .