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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

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

36. 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.

37. Example • Graph the following.

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

39. 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.

40. Examples • Graph the following.

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

42. Assignment • Ex. 2.2 (p. 64) #1-4

43. 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.

44. 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.

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

46. 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

47. Examples

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