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) = -x 2 + 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 companys 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)?