2. More on Functions & Graphs 2.2 JMerrill, 2007 Contributions by DDillon Revised 2008

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

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

5. Difference Quotient  For f(x) = x 2 – 4x + 7, find

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

7. x y 4 -4 A piecewise-defined function is composed of two or more functions. Piecewise-Defined Functions f(x) = 3 + x, x < 0 x 2 + 1, x 0 Use when the value of x is less than 0. Use when the value of x is greater or equal to 0. (0 is not included.) open circle (0 is included.) closed circle

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

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

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

11. decreasing on an interval if, for any x 1 and x 2 in the interval, x 1 f (x 2 ), constant on an interval if, for any x 1 and x 2 in the interval, f (x 1 ) = f (x 2 ). The graph of y = f (x): increases on ( – ∞, – 3), decreases on ( – 3, 3), increases on (3, ∞). Increasing, Decreasing, and Constant Functions A function f is: increasing on an interval if, for any x 1 and x 2 in the interval, x 1 < x 2 implies f (x 1 ) < f (x 2 ), (3, – 4) x y ( – 3, 6) P206

12. Function Extrema (or local)

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

14. Application During a 24-hour period, the temperature y (in degrees Fahrenheit) of a certain city can be approximated by the model y = 0.026x 3 – 1.03x 2 + 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)

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

16. A Function f is even if for each x in the domain of f, f (– x) = f (x). Even Functions x y f (x) = x 2 An even function is symmetric about the y-axis.

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

18. A Function f is odd if for each x in the domain of f, f (– x) = – f (x). Odd Functions x y f (x) = x 3 An odd function is symmetric with respect to the origin.

19. Summary of Even and Odd Functions & Symmetry 1.Replace x with –x 2.Simplify 3.If nothing changes, the function is even. If everything changes, the function is odd.

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