CHAPTER TWO: LINEAR EQUATIONS AND FUNCTIONS ALGEBRA TWO Section Piecewise Functions

LEARNING GOALS Goal One - Represent piecewise functions. Goal Two - Use piecewise functions to model real-life quantities.

VOCABULARY Piecewise functions Piecewise functions are represented by a combination of equations, each corresponding to a part of the function. step-function A step-function has a graph which resembles a set of stair steps. An example of a step function is the greatest integer function. This function is denoted by g(x) = [[x]], where for every real number x, g(x) is the greatest integer less than or equal to x.

Evaluating a Piecewise Function 1. Find which equation to used based on the equations' limits 2. Plug x into the correct equation and solve for f(x)

PROBLEM: Evaluate f (x) when (a) x = -1, (b) x = 1, and (c) x = 3. SOLUTION Evaluating a Piecewise Function f (x) = 2x + 3, if x < 0 f (x) = 2, if 0 < x < 2 f (x) = 2, if 0 < x < 2 f (x) = -x + 1, if x > 2 Since x = -1 is less than zero, use the first equation. f (x) = 2x + 3 f (x) = 2(-1) + 3 f (x) = 1

PROBLEM: Evaluate f (x) when (a) x = -1, (b) x = 1, and (c) x = 3. SOLUTION Evaluating a Piecewise Function f (x) = 2x + 3, if x < 0 f (x) = 2, if 0 < x < 2 f (x) = 2, if 0 < x < 2 f (x) = -x + 1, if x > 2 Since x = 1 is more than zero, use the second equation. f (x) = 2x + 3 f (-1) = 2(-1) + 3 f (-1) = 1 f (x) = 2 f (1) = 2

PROBLEM: Evaluate f (x) when (a) x = -1, (b) x = 1, and (c) x = 3. SOLUTION Evaluating a Piecewise Function f (x) = 2x + 3, if x < 0 f (x) = 2, if 0 < x < 2 f (x) = 2, if 0 < x < 2 f (x) = -x + 1, if x > 2 Since x = 3 is more than two, use the third equation. f (x) = 2x + 3 f (-1) = 2(-1) + 3 f (-1) = 1 f (x) = 2 f (1) = 2 f (x) = -x + 1 f (3) = -(3) + 1 f (3) = -2

PROBLEM: Evaluate f (x) when (a) x = 5, (b) x = 0, and (c) x = -2. SOLUTION Evaluating a Piecewise Function f (x) = x + 1, if x > 1 f (x) = -x - 2, if x < 1 Since x = 5 is more than one, use the first equation. f (x) = x + 1 f (5) = (5) + 1 f (5) = 6

PROBLEM: Evaluate f (x) when (a) x = 5, (b) x = 0, and (c) x = -2. SOLUTION Evaluating a Piecewise Function f (x) = x + 1, if x > 1 f (x) = -x - 2, if x < 1 Since x = 0 is less than one, use the second equation. f (x) = x + 1 f (5) = (5) + 1 f (5) = 6 f (x) = -x - 2 f (0) = -(0) - 2 f (0) = - 2

PROBLEM: Evaluate f (x) when (a) x = 5, (b) x = 0, and (c) x = -2. SOLUTION Evaluating a Piecewise Function f (x) = x + 1, if x > 1 f (x) = -x - 2, if x < 1 Since x = -2 is less than one, use the second equation. f (x) = x + 1 f (5) = (5) + 1 f (5) = 6 f (x) = -x - 2 f (0) = -(0) - 2 f (0) = - 2 f (x) = -x - 2 f (-2) = -(-2) - 2 f (-2) = 0

Graphing Piecewise Functions 1. Find the points where x changes 2. Graph the function in the appropriate areas 3. Check to make sure you draw open dot for, and closed dot for Graphing Piecewise Functions 1. Find the points where x changes 2. Graph the function in the appropriate areas 3. Check to make sure you draw open dot for, and closed dot for

PROBLEM: Graph the function f (x) = -x if x < 3 f (x) = (2/3)x - 4 if x > 3 Graphing a Piecewise Function To the right of x = 3, the graph is given by y = (2/3)x - 4. To the left of x = 3, the graph is given by y = -x. 1. Find the points where x changes 2. Graph the function in the appropriate areas 3. Check to make sure you draw open dot for, and closed dot for

PROBLEM: Graph the function f (x) = x + 2 if x > 1 f (x) = -x + 2 if x < 1 Graphing a Piecewise Function To the right of x = 1, the graph is given by y = x + 2. To the left of x = 1, the graph is given by y = -x + 2.

Graphing a Step Function PROBLEM: Graph the function f (x) = -2 if 0 < x < 2 f (x) = -4 if 2 < x < 4 f (x) = -6 if 4 < x < 6 The graph is composed of three line segments, because the function has three parts. The intervals of x tell you that each line segment is 2 units in length and begins with a solid dot and ends with an open dot.

Graphing a Step Function PROBLEM: Graph the function f (x) = -2 if 0 < x < 2 f (x) = -4 if 2 < x < 4 f (x) = -6 if 4 < x < 6

Graphing a Step Function PROBLEM: Graph the function f (x) = -2 if 0 < x < 2 f (x) = -4 if 2 < x < 4 f (x) = -6 if 4 < x < 6

Graphing a Step Function PROBLEM: Graph the function f (x) = -2 if 0 < x < 2 f (x) = -4 if 2 < x < 4 f (x) = -6 if 4 < x < 6

ASSIGNMENT pg #13, #17, #21, #25, #27, #35