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

2. 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

3. 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

4. 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

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

6. 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.

7. 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

8. 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

9. 2.6 – Special Functions Absolute Value function f(x) = |x| f(x) = |x|-2 2 -1 1

10. 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

11. 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.

12. 2.6 – Special Functions Example 3 Graph f(x) = {x – 1 if x ≤ 3
Identify the domain and range

13. 2.6 – Special Functions
Step function
Constant function

14. 2.6 – Special Functions
Absolute Value function
Piecewise function

15. 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

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