1. Lesson 1.3, page 154 More on Functions
Objectives
To find the difference quotient.
Understand and use piecewise functions
Identify intervals on which a function increases, decreases, or is constant.
Use graphs to locate relative maxima or minima.
Identify even or odd functions & recognize the symmetries.
Graph step functions.

2. REVIEW of Lesson 1.2 Reminder: Domain Restrictions
For FRACTIONS:
No zero in denominator!
For EVEN ROOTS:
No negative under even root!

3. Find the domain of each (algebraically)and write in interval notation.

4. Functions & Difference Quotients
Useful in discussing the rate of change of function over a period of time
EXTREMELY important in calculus
(h represents the difference in two x values)
DIFFERENCE QUOTIENT FORMULA:

5. Difference Quotient The average rate of change (the slope of the secant line)

6. If f(x) = -2x2 + x + 5, find and simplify each expression.
A) f(x+h)

7. If f(x) = -2x2 + x + 5, find and simplify each expression.B)

8. Your turn: Find the difference quotient: f(x) = 2x2 – 2x + 1

9. PIECEWISE FUNCTIONS
Piecewise function – A function that is defined differently for different parts of the domain; a function composed of different “pieces”
Note: Each piece is like a separate function with its own domain values.
Examples: You are paid $10/hr for work up to 40 hrs/wk and then time and a half for overtime.

10. See Example 3, page 169. Check Point 2: Use the function
to find and interpret each of the folllowing:
a) C(40) b) C(80)

11. Graphing Piecewise Functions
Draw the first graph on the coordinate plane.
Be sure to note where the inequality starts and stops. (the interval)
Erase any part of the graph that isn’t within that interval.

12. Graph
See p.1015
for more problems.

13. Describing the Function
A function is described by intervals, using its domain, in terms of x-values.
Remember:

14. Increasing and Decreasing Functions
Increasing: Graph goes “up” as you move from left to right.
Decreasing: Graph goes “down” as you move from left to right.
Constant: Graph remains horizontal as you move from left to right.

15. Increasing and Decreasing

16. Constant

17. Increasing and Decreasing

18. See Example 1, page 166.
Check Point 1 – See middle of
page 166.

19. Find the Intervals on the Domain in which the Function is Increasing, Decreasing, and/or Constant

20. Relative Maxima and Minima
based on “y” values
maximum – “peak” or highest value
minimum – “valley” or lowest value

21. Relative Maxima and Relative Minima

22. Even & Odd Functions & Symmetry
Even functions are those that are mirrored through the y-axis. (If –x replaces x, the y value remains the same.) (i.e. 1st quadrant reflects into the 2nd quadrant)
Odd functions are those that are mirrored through the origin. (If –x replaces x, the y value becomes –y.) (i.e. 1st quadrant reflects into the 3rd quadrant or over the origin)

23. See Example 2, page 167.
Determine whether each function is even, odd, or neither.
a) f(x) = x b) g(x) = 7x3 - x

24. Determine whether each function is even, odd, or neither.
c) h(x) = x5 + 1

25. Your turn: Determine if the function is even, odd, or neither.