1. B. Functions Calculus 30 C30.1 Extend understanding of functions including: algebraic functions (polynomial, rational, power) transcendental functions (exponential, logarithmic, trigonometric) piecewise functions, including absolute value.

2. 1. Introduction • C30.1 • Extend understanding of functions including: algebraic functions (polynomial, rational, power) transcendental functions (exponential, logarithmic, trigonometric) piecewise functions, including absolute value.

3. 1. Introduction • A relation is simply a set of ordered pairs. • A function is a set of ordered pairs in which each x-value is paired with one and only one y-value. • Graphically, we say that the vertical line test works.

4. can be written as • You perform a function on x, in this case you square it to get y. • So f(4)=16, f(-4)=16, f(3)=9, etc. • Notice no x’s are repeated , so this is a function.

5. The x-value, which can vary, is called the independent variable, and the y-value, which is determined from “doing something” to x is called the dependent variable • Functions can be represented in words: (square x to get y) • in a table of values: • in function notation: • Or on a graph:

6. Note*

7. We use “function notation” to substitute an x-value into an equation and find its y-value

8. Examples 1. For , find: • f(-3) • f(21) • f(w+4) • 3f(5)

9. Assignment • Ex. 2.1 (p. 55) #1-10

10. 2. Identifying Functions • C30.1 • Extend understanding of functions including: algebraic functions (polynomial, rational, power) transcendental functions (exponential, logarithmic, trigonometric) piecewise functions, including absolute value.

11. 2. Identifying Functions a) Polynomial Functions • n is a nonnegative integer and , , etc. are coefficients

12. Example 1. For , find the leading coefficient and the degree.

13. The polynomials function has degree “n” (the largest power) and leading coefficient

14. Example 2. For , find the leading coefficient and the degree.

15. A polynomial function of degree 0 are called constant functions and can be written f(x)=b • Slope = zero

16. Example 1.

17. Polynomial functions of degree 1 are called linear functions and can be written • y = mx + b • m= slope • b= y-intercept • Example Graph

18. The linear function is also called the identity function • Example graph

19. Polynomial functions of a degree 2 are called quadratic functions and can we written • Example graph

20. Polynomial functions of degree 3 are called cubic functions • Example Graph • Degree 4 functions with a negative leading coefficient • Example Graph • Degree 5 functions with a negative leading coefficient • Example graph

21. Summary: • Polynomial functions of an odd degree and positive leading coefficient begin in quadrant 3 and end in quadrant 1 • Polynomial functions of an odd degree and negative leading coefficient begin in quadrant 2 and end in quadrant 4 • Polynomial functions of an even degree and positive leading coefficient begin in quadrant 2 and end in quadrant 1 • Polynomial functions of an even degree and negative leading coefficient begin in quadrant 3 and end in quadrant 4

22. b) A Power Function can be written: • where n is a real number

23. If “n” is a positive integer, the power function is also a polynomial function • Examples

24. Examples • Graph the following on your graphing calculator: etc.

25. Notice that all the graphs pass through the points (0,0) and (1,1). • This is true for all power functions

26. If the power is and n is a positive integer >1, it is called a root function

27. Graph the following and find the interval for each.

28. If the power is negative, it is called a reciprocal function and can be written: • Its graph is an hyperbola with x and y axes as asymptotes. • Example Graph

29. c) A Rational Function is the ratio of 2 polynomial functions and can be written: Note*: the reciprocal function is also a rational function.

30. Any x-value which makes the denominator = 0 is a vertical asymptote. • If degree of p(x) < degree of q(x), there is a horizontal asymptote at y=0 (x-axis) • If degree of p(x) = degree of q(x), there is a horizontal asymptote at y = k, where k is the ratio of the leading coefficients of p(x) and q(x) respectively.

31. Example • Find the asymptotes of the following function.

32. d) An Algebraic Function is formed by performing a finite number of algebraic operations (such as with polynomials • Thus all rational functions are also algebraic functions.

33. Examples Using your graphing calculators graph the following:

34. Thus the graphs of algebraic functions vary widely.

35. e) Trig Functions – contain sin, cos, tan, csc, sec or cot. Examples

36. f) Exponential Functions have “x” as the exponent (rather than as the base, as in power functions) and can be written: • where b>0,

37. Graphs of exponential functions always pass through (0,1) and lie entirely in quadrants 1 and 2 • If b>1, the graph is always increasing and if 0<b<1, the graph is always decreasing. The x axis is a horizontal asymptote line.

38. Example • Graph the following.

39. g) Logarithmic Functions have “y” as the exponent and can be written • where b>0,

40. Graphs of log functions always pass through (1,0) and lie in quadrants 1 and 4 • If b>1, the graph is always increasing and if 0<b<1, the graph is always decreasing. The y-axis is a vertical asymptote line.

41. Examples • Graph the following.

42. h) Transcendental Functions are functions that are not algebraic. They included the trig functions, exponential functions, and the log functions

43. Assignment • Ex. 2.2 (p. 64) #1-4

44. 3. Piecewise and Step Function • C30.1 • Extend understanding of functions including: algebraic functions (polynomial, rational, power) transcendental functions (exponential, logarithmic, trigonometric) piecewise functions, including absolute value.

45. 3. Piecewise and Step Function a) A Piecewise Function is one that uses different function rules for different parts of the domain. • Watch open and closed intervals and use corresponding dots • To find values for the function, use the equation that contains that value (on the graph) in its domain.

46. Example • Graph • Find: • f(-11) • f(7) • f(0)

47. The Absolute Value Function is a piecework-defined function: • Graph

48. b) The graph of a step function looks like a series of steps. • The greatest integer function names the greatest integer that is less than or equal to x and is written

49. Examples

50. This function is also called the floor function because the function rounds non-integer values down. • The notation, is also used