1. Concept

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

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

4. Example 2A 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.

5. Example 2B Determine Properties of Reciprocal Functions 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. B. Identify the asymptotes, domain, and range of the function.

6. Example 2B 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.

7. Example 3B 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. This represents a transformation of the graph of B. Graph the function State the domain and range.

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

9. Example 3A A. Graph the function A.B. C.D.

10. Example 3B 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} B. State the domain and range of

11. Example 4A Write Equations A. COMMUTING A 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 r= dOriginal equation. t Solve the formula r = d for t. t

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

13. Example 4B Write Equations B. COMMUTING A 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.