Lesson 5

Lesson 5: Multiplying and Dividing Functions

Focus

As a jet flies, the rate that the plane uses fuel changes. One of the main factors for the change is the amount of fuel carried. Fuel is used up as the jet flies, so the jet gets lighter and uses less fuel. Finding a way to determine the average rate that fuel is used from the beginning of a flight to a given point in the flight can be useful when comparing aircraft efficiency.

In this lesson you will explore how functions can be multiplied and divided and how this information can be used to solve problems.

Lesson Outcomes

At the end of this lesson you will be able to

• multiply and divide functions
• determine the domain and range of the product or quotient of functions

Lesson Question

You will investigate the following question:

• How can functions be multiplied or divided?

Assessment

Your assessment may be based on a combination of the following tasks:

• completion of the Lesson 5 Assignment (Download the Lesson 5 Assignment and save it in your course folder now.)
• course folder submissions from Try This and Share activities
• additions to Formula Sheet

Discover

Try This 1

Consider the functions f(x) = x + 1 and g(x) = −x + 3.

1. 
   
   1. Predict what the graph of p(x) = (fg)(x) will look like. Explain how you made this prediction.
   2. Predict what the graph of will look like. Explain how you made this prediction.
2.  
   1. Use Multiply and Divide Functions to graph p(x) and q(x) and to check your predictions. Save screen captures of your two graphs.
      
      
       
      
   2. Explain how the graph of p(x) is related to the graphs of f(x) and g(x).
   3. Explain how the graph of q(x) is related to the graphs of f(x) and g(x).
3.  
   1. Use the equations of f(x) and g(x) to write an equation for p(x) and for q(x) in terms of x. Explain how you determined equations for p(x) and q(x).
   2. Are there any restrictions on x for p(x) or q(x)? Explain.
   3. Do the graphs of the equations you determined match the graphs from Multiply and Divide Functions? Should they?
4. Can the domain of p(x) or q(x) be determined from f(x)and g(x)? If so, explain how.

Save your responses in your course folder.

Share 1

With a partner or group, discuss the following questions based on the information in Try This 1.

1. How are multiplying and dividing functions similar? How are they different?
2. Compare multiplying and dividing functions to adding and subtracting functions. Describe both similarities and differences.

If required, save a record of your discussion in your course folder.

Explore

Multiplying and dividing functions is a straightforward process that is similar to adding or subtracting functions. In Try This 1 you found p(x) = (fg)(x) and Your calculations and domain explanations may have been similar to what is shown.

Determining the range of p(x) = (fg)(x) or g(x) ≠ 0 can be more difficult. This can often be interpreted from the graph of the function.

Interpreting graphs when multiplying or dividing functions is similar to interpreting graphs when adding or subtracting functions. Try This 2 explores this idea further.

Try This 2

Open Lesson 5 Printable Template. Either make a sketch of the graph or print the graph to complete the following activity.

1. 
   
   1. Use the graph to sketch h(x) = (fg)(x). Explain the procedure you used.
   2. Predict where any vertical asymptotes for and will occur. How are these asymptotes related to the non-permissible values of p(x) and q(x)?
   3. Sketch p(x) and q(x).
      
2. 
   
   1. What type of functions are h(x), p(x), and q(x)? Explain.
   2. State the domain for h(x), p(x), and q(x).

Save your responses in your course folder.

In Try This 2 you may have found that multiplying and dividing functions using their graphs follows a process similar to that of adding and subtracting functions using their graphs. Again, you can pick a point on each function with the same x-value and multiply or divide the y-values. Dividing Functions shows how to determine the graph of from Try This 2.

In Try This 2 you also saw that it is possible to predict the type of function produced by a multiplication or division of functions:

• h(x) is a quadratic polynomial function because it is the product of two linear functions.
• p(x) is a rational function because it is the quotient of two linear functions.
• q(x) is a rational function because it is the quotient of two linear functions.

Complete questions 2, 3, 4.a., 5.b., and 9 on page 496 of the textbook. Answer

In Try This 3 you will apply your understanding of multiplying and dividing functions to a problem that involves jet fuel.

Try This 3

Suppose that the volume of fuel a jet has used from the time it reaches its cruising altitude to time t is given by the function f(t) = 15 500t − 250t2, where f(t) is measured in litres and t is in hours. The distance the plane has travelled t hours after reaching its cruising altitude is d(t) = 900t, where d(t) is measured in kilometres.

1. Determine a function, r(t), that gives the average rate of fuel use per kilometre for the cruising portion of the trip after t hours.
2. If the cruising portion of the trip is expected to take 15 h, determine the domain and range of f(t), d(t), and r(t).
3. 1. Determine the average rate of fuel use per kilometre after 3 h and after 12 h.
   2. Explain the difference seen in question a.
4. At what time is the average rate of fuel use 15.4 L/km?

Save your responses in your course folder.

Share 2

With a partner or group, discuss the following questions based on the information in Try This 3.

1. Why is r(t) meaningless when t = 0 in terms of the scenario?
2. Why would question 4 of Try This 3 have been difficult if you didn’t use the quotient function r(t)?

If required, save a record of your discussion in your course folder.

Connect

Lesson 5 Assignment

Lesson 5 Summary

In this lesson you saw that it is possible to add and subtract functions. The notation h(x) = (fg)(x) is equivalent to h(x) = (fx)g(x), and is equivalent to To multiply functions using their graphs, multiply the y-coordinates for any x-coordinate. Similarly, to divide functions using their graphs, divide the y-coordinates for any x-coordinate.

When dividing functions, remember that there is a restriction on the domain where the function in the denominator equals zero. The range of the combined function can be determined from the graph of the combined function.

So far, you have usually seen a number or variable as the input for a function. In Lesson 6 you will explore how a function can be the input for a function.