Lesson 1.3, page 154 More on Functions

1. Lesson 1.3, page 154More 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 1.2Reminder: 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 QuotientThe 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) = x2 + 6 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. • Even • Odd • Neither