8.4 An Introduction to Functions: Linear Functions,

Applications, and Models

Part 1: Functions

Relations

• Often we describe one quantity in terms of another, using ordered pairs.– When you fill your tank with gas, the total amount you pay

is equal to the number of gallons multiplied by the price per gallon.

• If the value of y depends on the value of x, then y is the dependent variable and x is the independent variable.– The “amount you pay” depends on the “number of

gallons”.• When quantities are related in this way, we call it a relation.– A relation is a set of ordered pairs.

Functions

• A function is a relation in which for each value of the first component of the ordered pairs there is exactly one value of the second component.– For each value of x there is only one value of y.

F = {(1, 2), (-2, 5), (3, -1)}G = {(-4, 1), (-2, 1), (-2, 0)}• Which is a relation?• Which is a function?

Function Notation

• When “y is a function of x”, we can use the notation f(x) which shows that y depends on x.– This is called function notation.– This does NOT mean multiply f and x!

• Note that f(x) is just another name for y.– y = 2x – 7 and f(x) = 2x – 7 are the same!

Using Function Notation

• Let f(x) = -x2 + 5x – 3. Find the following:

f(2)

f(-1)

f(2x)

Linear Functions

• A function that can be written in the form f(x) = mx + b

for real numbers m and b is a linear function.

– This is the same as y = mx + b!

Graphing Linear Functions

• Graph the linear function f(x) = -2x + 3.

Graph the linear function f(x) = 3.

Modeling with Linear Functions

• A company’s cost of producing a product and the revenue from selling the product can be expressed as linear functions.

• The idea of break-even analysis then can be explained using the graphs of these functions.– When the cost equals the revenue, the company breaks

even.– When cost is greater than revenue, the company loses

money.– When the cost is less than revenue, the company makes

money.

Analyzing Cost, Revenue, and Profit

• A company that produces DVDs of live concerts places an ad in a newspaper. The cost of the ad is $100. Each DVD costs $20 to produce and is sold for $24.

Express the cost C as a function of x, the number of DVDs produced.

Express the revenue R as a function of x, the number of DVDs sold.

When will the company break even (what value of x makes revenue equal cost)?