Trig/Pre-Calculus Opening Activity Write the domain of the following functions. Solve the following inequalities.

(2, –2) is on the graph of f(x) = (x – 1)2 – 3. The graph of a function f is the collection of ordered pairs (x, f(x)) where x is in the domain of f. (2, –2) is on the graph of f(x) = (x – 1)2 – 3. x y 4 -4 f(2) = (2 – 1)2 – 3 = 12 – 3 = – 2 (2, –2) Definition of Graph

The domain of the function y = f (x) is the set of values of x for which a corresponding value of y exists. The range of the function y = f (x) is the set of values of y which correspond to the values of x in the domain. x y 4 -4 Range Domain Domain & Range

Example: Domain & Range Example: Find the domain and range of the function f (x) = from its graph. x y – 1 1 Range (–3, 0) Domain The domain is [–3,∞). The range is [0,∞). Example: Domain & Range

Increasing, Decreasing, and Constant Functions A function f is: increasing on an interval if, for any x1 and x2 in the interval, x1 < x2 implies f (x1) < f (x2), decreasing on an interval if, for any x1 and x2 in the interval, x1 < x2 implies f (x1) > f (x2), constant on an interval if, for any x1 and x2 in the interval, f (x1) = f (x2). (3, – 4) x y (–3, 6) –2 2 The graph of y = f (x): increases on (– ∞, –3), decreases on (–3, 3), increases on (3, ∞). Increasing, Decreasing, and Constant Functions

Minimum and Maximum Values A function value f(a) is called a relative minimum of f if there is an interval (x1, x2) that contains a such that x1 < x < x2 implies f(a) f(x). x y Relative maximum Relative minimum A function value f(a) is called a relative maximum of f if there is an interval (x1, x2) that contains a such that x1 < x < x2 implies f(a) f(x). Minimum and Maximum Values

Graphing Utility: Approximating a Relative Minimum Graphing Utility: Approximate the relative minimum of the function f(x) = 3x2 – 2x – 1. – 6 6 – 0.86 – 4.79 – 1.79 2.14 Zoom In: 0.58 0.76 -3.24 -3.43 Zoom In: The approximate minimum is (0.67, –3.33). Graphing Utility: Approximating a Relative Minimum

Determine the relative minima and maxima of the following function Determine the relative minima and maxima of the following function. Determine where the graph is increasing, decreasing, and constant.

Piecewise-Defined Functions A piecewise-defined function is composed of two or more functions. f(x) = 3 + x, x < 0 x2 + 1, x 0 Use when the value of x is less than 0. Use when the value of x is greater or equal to 0. x y 4 -4 open circle closed circle (0 is not included.) (0 is included.) Piecewise-Defined Functions

f (x) = x2 is an even function. A function f is even if for each x in the domain of f, f (– x) = f (x). Symmetric with respect to the y-axis. x y f (x) = x2 f (– x) = (– x)2 = x2 = f (x) f (x) = x2 is an even function. Even Functions

f (x) = x3 is an odd function. A function f is odd if for each x in the domain of f, f (– x) = – f (x). f (x) = x3 x y f (– x) = (– x)3 = –x3 = – f (x) Symmetric with respect to the origin. f (x) = x3 is an odd function. Odd Functions

Now we are going to graph the piecewise function from DNA #4-6 by HAND.

This graph does not pass the vertical line test. It is not a function. A relation is a function if no vertical line intersects its graph in more than one point. x y 4 -4 x y 4 -4 x = | y – 2| y = x – 1 This graph does not pass the vertical line test. It is not a function. This graph passes the vertical line test. It is a function. Vertical Line Test

Graph this…

Increasing, Decreasing, and Constant Functions Consider…

Relative Minimum and Maximum Values. We will use a graphing utility to find the following functions relative minima and maxima.

EVEN Functions ODD Functions Every EVEN function is symmetric about the y-axis. Every ODD function is symmetric about the y-axis.

Now we are going to graph the piecewise function from DNA #4-6 by HAND.

Graph this…

Ex 4) The net sales for a car manufacturer were $14.61 billion in 2005 and $15.78 billion in 2006. Write a linear equation giving the net sales y in terms of x, where x is the number of years since 2000. Then use the equation to predict the net sales for 2007.

Graph the following linear functions Graph the following linear functions. Graph #1 – 3 on the same coordinate plane.