More on Functions & Graphs 2.2

1. More on Functions & Graphs2.2 JMerrill, 2007 Contributions by DDillon Revised 2008

2. Review • Find: • Domain • [-1, 4) • Range • [-5, 4] • f(-1) • f(-1) = -5 • f(2) • f(2) = 4

3. Difference Quotient • One of the basic definitions in calculus uses the difference quotient ratio: • It applies to average rate of change.

4. Difference Quotient • For f(x) = x2 – 4x + 7, find

5. Difference Quotient You Do • Given f(x) = 3x – 1, find • 3

6. 3 + x, x < 0 f(x) = x2 + 1, x 0 y x 4 -4 Piecewise-Defined Functions A piecewise-defined function is composed of two or more functions. Use when the value of x is less than 0. Use when the value of x is greater or equal to 0. opencircle closed circle (0 is not included.) (0 is included.)

7. Evaluating A Piecewise-Defined Function • Evaluate the function when x = -1 and x = 0 • When x = -1, that is less than 0, so you only use the top function • f(-1) = (-1)2 + 1 = 2 • When x = 0, use the bottom function • f(0) = 0 – 1 = -1

8. You Do • Solve • A. f(-1) • B. f(0) • C. f(2) f(-1) = -1 f(0) = 2 f(2) = 6

9. y (–3, 6) x (3, – 4) Increasing, Decreasing, and Constant Functions Where is this function increasing? Where is it decreasing? The graph of y = f(x): • increases on (–∞, –3), • decreases on (–3, 3), • increases on (3, ∞).

10. y (–3, 6) x (3, – 4) Increasing, Decreasing, and Constant Functions A function f is: • increasing on an interval if, for any x1 and x2 in the interval, x1 < x2 implies f(x1) < f(x2), • decreasing on an interval if, for any x1 and x2 in the interval, x1 < x2 implies f(x1) > f(x2), P206 • constant on an interval if, for any x1 and x2 in the • interval, f(x1) = f(x2). The graph of y = f(x): • increases on (–∞, –3), • decreases on (–3, 3), • increases on (3, ∞).

11. Function Extrema (or local) (or local)

12. Find Extrema and Intervals of Increasing and Decreasing Behavior. y = x3 – 3x Relative max exists at -1. Relative max = 2 Relative min is exists at 1. Relative min = -2

13. Application During a 24-hour period, the temperature y (in degrees Fahrenheit) of a certain city can be approximated by the model y = 0.026x3 – 1.03x2 + 10.2x + 34, 0 ≤ x ≤ 24, where x represents the time of day, with x = 0 corresponding to 6 AM. Approximate the maximum and minimum temperatures during this 24-hour period. Maximum: about 64°F (at 12:36 PM) Minimum: about 34°F (at 1:48 AM)

14. Even Functions A Function f is even if for each x in the domain of f, f(–x) = f(x). f(x) = x2 f(–x) = (–x)2=x2 If you get the same thing you started with, it is an even function f(x) = x2 is an even function.

15. y x Even Functions A Function f is even if for each x in the domain of f, f(–x) = f(x). An even function is symmetric about the y-axis. f(x) = x2

16. Odd Functions A Function f is odd if for each x in the domain of f, f(–x) = –f(x). f(x) = x3 f(–x) = (–x)3=–x3 If all terms change signs the function is odd. f(x) = x3 is an odd function.

17. y x Odd Functions A Function f is odd if for each x in the domain of f, f(–x) = –f(x). f(x) = x3 An odd function is symmetric with respect to the origin.

18. Summary of Even and Odd Functions & Symmetry • Replace x with –x • Simplify • If nothing changes, the • function is even. If • everything changes, the • function is odd.

19. Even, Odd, or Neither? f(x) = x3 + 2 Check f(-x) f(-x) = (-x)3 + 2 f(-x) = -x3 + 2 Not even, because not equal to f(x). Not odd, because not equal to –f(x). This function is neither even nor odd.