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Lesson

Before beginning this Investigation, use your knowledge of composition of functions to answer the "Think about ... " located in the margin on page 132 of your text. You may check your response when you have completed the question. 

The graphs in the Investigation have been carefully chosen to lead you to discover some specific properties about the relationships between the graphs of functions and the graphs of their reciprocals. 

As you progress through this activity, you should be prompted to think about other kinds of reciprocal functions, besides transformations of the reciprocal function of the function y = x.

Begin the Investigation, completing Step A only at this time. 

After drawing the graphs indicated, you should have a good idea about the notion of using the graph of one function to get the graph of its reciprocal. 

  • Are you able to make any other conjectures? 
  • Do you notice any relationship between the zeros of a given function and the vertical asymptote for its reciprocal? 
  • Can you predict the vertical asymptotes for the reciprocal of the graph of
     f    (x) = x2 - 1? For the reciprocal of f(x) = x2 +1? 
  • As the graph of y = x2 gets higher, why does the graph of get lower? Do you think this graph will ever hit the x-axis? Why or why not? 

After giving the above questions careful consideration, you may proceed with the rest of the Investigation and related questions. 

Once completed, you may view a summary of the Investigation. As well, more questions are provided for you to answer.

Two examples, similar to the Check Your Understanding questions, are provided below.

Activity

C.Y.U. pages 134 & 135 #'s 19 - 21

When you have completed these questions, ask your on-site teacher to get the solutions for you from the Teacher's Resource Binder and check them against your answers. After you do this, if there is something you had trouble with and still do not understand, contact your on-line teacher for help.

Test Yourself

  1. Given the function , state the domain, range, the equations of any asymptotes, and the intervals of increase and decrease.
  2. What would be the equation of the vertical asymptote(s) for the graphs of the reciprocal of the following functions?
    (a) f(x) = x - 8       (b) f(x) = x2 - 49       (c) f(x) = x2 + 9

Solutions

1.

  • Domain:
  • Range:
  • Vertical asymptote: x = 9
  • Horizontal asymptote: y = 0
  • Increases on (-, 9)
  • Decreases on (9, )

2. (a) x = 8     (b) x = 7     (c) no vertical asymptotes