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Section 3.6 Reciprocal Functions

Section 3.6 Reciprocal Functions. Objectives: 1. To identify vertical asymptotes, domains, and ranges of reciprocal functions. 2. To graph reciprocal functions. Definition. Reciprocal Function Any function that is a reciprocal of another function. 1. sec. x. =. cos x. 1. csc.

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Section 3.6 Reciprocal Functions

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  1. Section 3.6 Reciprocal Functions

  2. Objectives: 1. To identify vertical asymptotes, domains, and ranges of reciprocal functions. 2. To graph reciprocal functions.

  3. Definition Reciprocal FunctionAny function that is a reciprocal of another function.

  4. 1 sec x = cos x 1 csc x = sin x 1 cot x = tan x Definition Reciprocal trigonometric ratios:

  5. Definition Reciprocal trigonometric functions: y = sec  y = csc  y = cot 

  6. These functions are examples of a larger class of reciprocal functions, including reciprocals of power, polynomial, and exponential functions.

  7. 1 3x4 1 x2 – 4 g(x) = f(x) = 1 4x h(x) = Examples of reciprocal functions k(x) = sec x

  8. 1 3x4 f(x) = 1 3(1)4 1 3 f(1) = = EXAMPLE 1Find f(1), g(2), h(1/2), and k(/4), using the functions above.

  9. 1 x2 – 4 g(x) = 1 22 – 4 1 0 g(2) = = , which is undefined EXAMPLE 1Find f(1), g(2), h(1/2), and k(/4), using the functions above.

  10. 1 2 1 41/2 1 4 h(1/2) = = = 1 4x h(x) = EXAMPLE 1Find f(1), g(2), h(1/2), and k(/4), using the functions above.

  11. 1 cos /4 1 2/2 k(/4) = sec /4 = = 2 = = 2 2 EXAMPLE 1Find f(1), g(2), h(1/2), and k(/4), using the functions above. k(x) = sec x

  12. Since reciprocal functions have denominators, you must be careful about what values are used in the domain.

  13. EXAMPLE 2Find the domains of f(x), g(x), h(x), and k(x) in the previous functions. Find all values for which the denominator of f(x) and g(x) equals zero. f(x) 3x4 = 0 x4 = 0 x = 0 g(x) x2 – 4 = 0 x2 = 4 x = ±2

  14. EXAMPLE 2Find the domains of f(x), g(x), h(x), and k(x) in the previous functions. Exclude those values from the domain. f(x): D = {x|x  R, x ≠ 0} g(x): D = {x|x  R, x ≠ ±2}

  15. EXAMPLE 2Find the domains of f(x), g(x), h(x), and k(x) in the previous functions. Since 4x ≠ 0  x, the domain of h(x) is R. Since cos x = 0 when x = /2 + k, k  R, the domain of k(x) is D = {x|x  R, x ≠ /2 + k, k  Z}.

  16. 1 x2 – 4 EXAMPLE 3Graph g(x) = . Give the domain and range. Is g(x) continuous? Is g(x) an odd or even function?

  17. 1 x2 – 4 EXAMPLE 3Graph g(x) = . Use reciprocal principles to graph g(x).

  18. 1 x2 – 4 EXAMPLE 3Graph g(x) = . Use reciprocal principles to graph g(x).

  19. 1 x2 – 4 EXAMPLE 3Graph g(x) = . D = {x|x ≠ ±2} R = {y|y  0 or y  -1/4} g(x) is an even function but is not continuous.

  20. 1 x2 – 4 EXAMPLE 4Graph g(x) = . again without graphing its reciprocal function first. 1. Find the domain excluding values where the denominator equals zero. x2 – 4 = 0 x2 = 4 x = ±2 D = {x|x ≠ ±2}

  21. 1 x2 – 4 EXAMPLE 4Graph g(x) = . again without graphing its reciprocal function first. 2. Check for x-intercepts. Since the numerator cannot equal zero, the graph cannot touch the x-axis.

  22. 1 x2 – 4 EXAMPLE 4Graph g(x) = . again without graphing its reciprocal function first. 3. Plot a point in each of the regions determined by the asymptotes (2 & -2). Since the graph cannot cross the x- axis, points within a region will all be on the same side of the x-axis. Include the y-intercept as one of the points.

  23. 1 x2 – 4 EXAMPLE 4Graph g(x) = . again without graphing its reciprocal function first. 4. Use the asymptotes as guides. Your graph will never quite reach either vertical asymptote or the x-axis.

  24. 1 x2 – 4 EXAMPLE 4Graph g(x) = .

  25. EXAMPLE 5Graph y = csc x. Give the domain, range, zeros, and period. Is it continuous?

  26. EXAMPLE 5Graph y = csc x. Give the domain, range, zeros, and period. Is it continuous? D = {x|x ≠ k, k  Z} R = {y|y  1 or y  -1} The function has no zeros; the period is 2. It is not continuous.

  27. Homework: pp. 148-151

  28. ►A. Exercises 1. f(x)

  29. ►A. Exercises 3. p(x)

  30. h(x) = 1 x2 + 5x – 14 ►A. Exercises 6. Give the vertical asymptotes of

  31. 1 x2 – 25 ►A. Exercises Evaluate each function as indicated. 9. f(x) = for x = 2 and x = -6

  32. h(x) = 1 x2 + 5x – 14 ►B. Exercises 12. Graph the reciprocal function. Give the domain and range.

  33. ■ Cumulative Review 41. Solve ABC where A = 58°, B = 39°, and a = 10.5.

  34. ■ Cumulative Review 42. Give the period and amplitude of y = 5 sin 3x.

  35. x – 8 if x  3 x2 – 1 if 3  x  9 7x if x  9    f(x) = ■ Cumulative Review 43. Find f(4) if

  36. ■ Cumulative Review 44. How many zeros does a cubic polynomial function have? Why?

  37. ■ Cumulative Review 45. Graph y = 2x and estimate 20.7 from the graph.

  38. A summary of principles for graphing reciprocal functions follows: 1. The larger the number, the closer the reciprocal is to zero. 2. The reciprocal of 1 and -1 is itself. 3. There is a vertical asymptote for the reciprocal when f(x) = 0.

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