Unit 1 – First-Degree Equations and Inequalities Chapter 2 – Linear Relations and Functions 2.6 – Special Functions

2.6 – Special Functions In this section we will learn about: Identifying and graphing step, constant, and identity functions Identify and graph absolute value and piecewise functions

Weight not over (ounces) 2.6 – Special Functions Step function – not linear, increases by step (line segments or rays) Cost of postage to mail a letter Letters between whole numbers, the cost “steps up” to the next higher cost Weight not over (ounces) Price ($) 1 0.39 2 0.63 3 0.87 4 1.11

2.6 – Special Functions Greatest integer function – example of a step function f(x) = [[x]] means greatest integer less than or equal to x [[7.3]] = 7 [[-1.5]] = -2 because -1 > -1.5

2.6 – Special Functions Greatest integer function (cont) f(x) = [[x]] -3 -2 ≤ x < -1 -2 -1 ≤ x < 0 -1 0 ≤ x < 1 1 ≤ x < 2 1 2 ≤ x < 3 2 3 ≤ x < 4 3

2.6 – Special Functions Example 1 One psychologist charges for counseling sessions at the rate of $85 per hour or any fraction thereof. Draw a graph that represents this situation.

2.6 – Special Functions Slope – intercept form: y = mx + b Function notation: f(x) = mx + b When m = 0, f(x) = b constant function Ex. f(x) = 3 f(x) = 0 is called the zero function

2.6 – Special Functions Another special case of slope-intercept form is when m = 1 and b = 0 f(x) = x Identity function Does not change the input value

2.6 – Special Functions Absolute Value function f(x) = |x| f(x) = |x| -2 2 -1 1

2.6 – Special Functions The absolute value function can be written as f(x) = {-x if x < 0 {x if x ≥ 0 A function that is written using two or more expressions is called a piecewise function

2.6 – Special Functions Example 2 Graph f(x) = |x – 3| and g(x) = |x + 2| on the same coordinate plane. Determine the similarities and differences in the two graphs.

2.6 – Special Functions Example 3 Graph f(x) = {x – 1 if x ≤ 3 Identify the domain and range

2.6 – Special Functions Step function Constant function

2.6 – Special Functions Absolute Value function Piecewise function

2.6 – Special Functions Example 4 Determine whether each graph represents a step function, a constant function, an absolute value function, or a piecewise function

HOMEWORK Page 99 #15 – 33 odd, 34 – 40 all 2.6 – Special Functions HOMEWORK Page 99 #15 – 33 odd, 34 – 40 all