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Understanding Reciprocal Functions: Domain, Range, and Asymptotes

Explore the properties of reciprocal functions, including determining the domain and range, identifying asymptotes, and understanding limitations. Examples and graph transformations are provided.

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Understanding Reciprocal Functions: Domain, Range, and Asymptotes

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  1. Splash Screen

  2. Concept

  3. Determine the values of x for which is not defined. Limitations on Domain Factor the denominator of the expression. Answer: The function is undefined for x = –8 and x = 3. Example 1

  4. Determine Properties of Reciprocal Functions A. Identify the asymptotes, domain, and range of the function. Identify the x-values for which f(x) is undefined. x – 2 = 0 x = 2 f(x) is not defined when x = 2. So, there is an asymptote at x = 2. Example 2A

  5. Determine Properties of Reciprocal Functions From x = 2, as x-values decrease, f(x)-values approach 0, and as x-values increase, f(x)-values approach 0. So, there is an asymptote at f(x) = 0. Answer: There are asymptotes at x = 2 and f(x) = 0. The domain is all real numbers not equal to 2 and the range is all real numbers not equal to 0. Example 2A

  6. Determine Properties of Reciprocal Functions B. Identify the asymptotes, domain, and range of the function. Identify the x-values for which f(x) is undefined. x + 2 = 0 x = –2 f(x) is not defined when x = –2. So, there is an asymptote at x = –2. Example 2B

  7. Determine Properties of Reciprocal Functions From x = –2, as x-values decrease, f(x)-values approach 1, and as x-values increase, f(x)-values approach 1. So, there is an asymptote at f(x) = 1. Answer: There are asymptotes at x = –2 and f(x) = 1. The domain is all real numbers not equal to –2 and the range is all real numbers not equal to 1. Example 2B

  8. A. Identify the asymptotes of the function. A.x = 3 and f(x) = 3 B.x = 0 and f(x) = –3 C.x = –3 and f(x) = –3 D.x = –3 and f(x) = 0 Example 2A

  9. B. Identify the domain and range of the function. • D = {x | x≠ –3}; R = {f(x) | f(x) ≠ –4} • B. D = {x | x≠ 3}; R = {f(x) | f(x) ≠ 0} • C. D = {x | x≠ 4}; R = {f(x) | f(x) ≠ –3} • D. D = {x | x≠ 0}; R = {f(x) | f(x) ≠ 4} Example 2B

  10. Concept

  11. A. Graph the function State the domain and range. This represents a transformation of the graph of Graph Transformations a = –1: The graph is reflected across the x-axis. h = –1: The graph is translated 1 unit left. There is an asymptote at x = –1. k = 3: The graph is translated 3 units up. There is an asymptote at f(x) = 3. Example 3A

  12. Graph Transformations Answer: Domain: {x│x ≠ –1}Range: {f(x)│f(x) ≠ 3} Example 3A

  13. B. Graph the function State the domain and range. This represents a transformation of the graph of Graph Transformations a = –4: The graph is stretched vertically and reflected across the x-axis. h = 2: The graph is translated 2 units right. There is an asymptote at x = 2. Example 3B

  14. Graph Transformations k = –1: The graph is translated 1 unit down. There is an asymptote at f(x) = –1. Answer: Domain: {x│x ≠ 2}Range: {f(x)│f(x) ≠ –1} Example 3B

  15. A. Graph the function • B. • C.D. Example 3A

  16. B. State the domain and range of A. Domain: {x│x ≠ –1}; Range: {f(x)│f(x) ≠ –2} B. Domain: {x│x ≠ 4}; Range: {f(x)│f(x) ≠ 2} C. Domain: {x│x ≠ 1}; Range: {f(x)│f(x) ≠ –2} D. Domain: {x│x ≠ –1}; Range: {f(x)│f(x) ≠ 2} Example 3B

  17. t Solve the formula r = d for t. t r = d Original equation. Write Equations A. COMMUTINGA commuter train has a nonstop service from one city to another, a distance of about 25 miles. Write an equation to represent the travel time between these two cities as a function of rail speed. Then graph the equation. Divide each side by r. d = 25 Example 4A

  18. Graph the equation Write Equations Answer: Example 4A

  19. Write Equations B. COMMUTINGA commuter train has a nonstop service from one city to another, a distance of about 25 miles. Explain any limitations to the range and domain in this situation. Answer:The range and domain are limited to all real numbers greater than 0 because negative values do not make sense. There will be further restrictions to the domain because the train has minimum and maximum speeds at which it can travel. Example 4B

  20. A.B. C.D. A. TRAVEL A commuter bus has a nonstop service from one city to another, a distance of about 76 miles. Write an equation to represent the travel time between these two cities as a function of rail speed. Example 4A

  21. A.B. C.D. B. TRAVELA commuter bus has a nonstop service from one city to another, a distance of about 76 miles. Graph the equation to represent the travel time between these two cities as a function of rail speed. Example 4

  22. Homework P. 549 # 2 – 34 even

  23. End of the Lesson

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