Section Functions Evaluate a Function Identify a Function Given By An Equation Find the Domain of a Function Graph of a Function Vertical Line Test for Functions Find the Domain and Range From a Graph Definition of Increasing, Decreasing, and Constant Functions Horizontal Line Test for a One-to-One Function Piecewise Functions Definition of the Greatest Integer Function

Evaluate each of the following for the function: f(x) = x² - 11 a. f(7)b. f(-5)c. f(3h)d. f(w + 3) a. f(7) = 7² - 11 = 49 – 11 = 38 b. f(-5) = (-5)² - 11 = 25 – 11 = 14 c. f(3h) = (3h)² - 11 = 9h² - 11 d. f(w + 3) = (w + 3)² - 11 = w² + 6w + 9 – 11 = w² + 6w - 2

How to Identify a Function Given By an Equation To determine whether an equation defines a function: 1. Solve (if possible) for the dependent variable, y. 2. Determine whether each single value of the independent variable produces exactly one value of the dependent variable. 3x + y = 1 y = -3x + 1 Since -3x + 1 is a unique real number for each value of x, this equation defines y as a function of x. -4x² + y² = 9 y² = 4x² + 9 y = + √4x² + 9 The right side of this equation produces two values of y for each value of x. It is not a function.

Graph of a Function To graph a function plot ordered pair solutions to the related function. To graph f(x) = x² - 1, think of the equation y = x² - 1. The domain of a function can be determined by observing the x-coordinates. The range can be determined by observing the y-coordinates.

The Vertical Line Test The definition of a function implies that any vertical line will intersect the graph at only one point. The Vertical Line Test A graph is the graph of a function if and only if no vertical line intersects the graph at more than one point.

Find the Domain and Range from a Graph Domain: The graph extends from x = 1 on the left to x = 9 on the right. So the domain is [1, 9] Range: The graph extends from a height of y = 2 to a height of y = 12. So the range is [2, 12]

Increasing, Decreasing and Constant Functions For values of x 3. We call these intervals increasing, constant, and decreasing.

Horizontal Line Test for a one-to- one Function If any horizontal line intersects the graph at most once, then the graph is the graph of a one-to-one function.

Piecewise Functions Piecewise functions are functions represented by more than one equation. To graph a piecewise function, graph each equation separately on the same graph. Graph the function: 2 if x < 1 x if x > 1

Greatest Integer Function For any real number, x, the greatest integer function f(x) = [[x]], is equal to x if x is an integer. It is equal to the greatest integer less than x if x is not an integer. [[3]] = ___[[2.72]] = ___ [[0.954]] = ___ [[-3.7]] = ____ When the greatest integer function is graphed it is known as a step function, because it resembles steps.

Solve an Application A car was purchased for$16,500. Assuming the car depreciates at a constant rate of $2200 per year (straight line depreciation) for the first seven years, write the value of the car, v, as a function of time, and calculate the value of the car after 3 years. t will represent the number of years after the car was purchased. Then 2200t is the amount that the car has depreciated after t years. So the value of the car written as a function of time will be: v(t) = 16,500 – 2200t After 3 years: v(3) =16,500 – 2200(3) $9900

Solve an Application A lighthouse is 2 miles south of a port. A ship leaves port and sails due east at a rate of 7 mph. Express the distance, d, between the ship and the lighthouse as a function of time, given that the ship has been sailing for t hours.

Assignment Page 138 – 142 # 1 – 9 odd, 13 – 37 odd,