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Multiplying and Dividing Functions

Author: Sophia
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what's covered
In this lesson, you will learn how to multiply or divide two functions. Specifically, this lesson will cover:

Table of Contents

1. Multiplying Two Functions

Oftentimes when working with functions, we will be asked to determine the result of multiplying two functions. When this occurs, there are two ways to approach the problem.

  • Method 1: Different Values of x If the two functions are in terms of the same variable BUT being evaluated for different values, we can evaluate each function for the value we are given and then multiply.
  • Method 2: Same Values of x If the two functions are in terms of the same variable and are being evaluated for the same value, we can multiply both functions together and substitute the value we are given to find a solution. Let's look at both methods.

1a. Method 1: Different Values of x

When given two functions evaluated at different values, evaluate each function separately then simply multiply the results together.

EXAMPLE

Find the value of f open parentheses 3 close parentheses g open parentheses 2 close parentheses when f open parentheses x close parentheses equals x squared minus 2 and g open parentheses x close parentheses equals x plus 7.

To solve this problem, first evaluate the functions f open parentheses x close parentheses when x equals 3 and g open parentheses x close parentheses when x equals 2, separately.

f open parentheses x close parentheses equals x squared minus 2 Find f open parentheses 3 close parentheses by substituting x with 3
f open parentheses 3 close parentheses equals open parentheses 3 close parentheses squared minus 2 Evaluate 3 squared
f open parentheses 3 close parentheses equals 9 minus 2 Subtract 2 from 9
f open parentheses 3 close parentheses equals 7 Our solution for f open parentheses 3 close parentheses
g open parentheses x close parentheses equals x plus 7 Find g open parentheses 2 close parentheses by substituting x with 2
g open parentheses 2 close parentheses equals 2 plus 7 Add 2 and 7
g open parentheses 2 close parentheses equals 9 Our solution for g open parentheses 2 close parentheses
f open parentheses 3 close parentheses g open parentheses 2 close parentheses equals 7 times 9 Find f open parentheses 3 close parentheses g open parentheses 2 close parentheses by multiplying 7 and 9
f open parentheses 3 close parentheses g open parentheses 2 close parentheses equals 63 Our solution

1b. Method 2: Same Values of x

When multiplying two functions given in terms of the same variable and evaluated at the same value we can multiply both equations first and then evaluate for the value we are given. Often times the notation used for this is f open parentheses a close parentheses g open parentheses a close parentheses equals open parentheses f times g close parentheses open parentheses a close parentheses.

EXAMPLE

Find the value of f open parentheses 3 close parentheses g open parentheses 3 close parentheses using the same functions given in Method 1.

table attributes columnalign left end attributes row cell f open parentheses x close parentheses equals x squared minus 2 end cell row cell g open parentheses x close parentheses equals x plus 7 end cell end table

Because each function is dependent on x and we are evaluating it for them for the same value, 3, we can multiply both functions first and then evaluate them for the given value. Let's look at how to do this.

f open parentheses x close parentheses g open parentheses x close parentheses Substitute f open parentheses x close parentheses and g open parentheses x close parentheses with their equivalent expressions
f open parentheses x close parentheses g open parentheses x close parentheses equals open parentheses x squared minus 2 close parentheses open parentheses x plus 7 close parentheses FOIL
f open parentheses x close parentheses g open parentheses x close parentheses equals x cubed plus 7 x squared minus 2 x minus 14 Plug in 3 for x
f open parentheses 3 close parentheses g open parentheses 3 close parentheses equals open parentheses 3 close parentheses cubed plus 7 open parentheses 3 close parentheses squared minus 2 open parentheses 3 close parentheses minus 14 Evaluate
f open parentheses 3 close parentheses g open parentheses 3 close parentheses equals 27 plus 63 minus 6 minus 14 Simplify
f open parentheses 3 close parentheses g open parentheses 3 close parentheses equals 70 Our solution

2. Dividing Two Functions

When we divide two functions we go through a process very similar to that used when multiplying two functions; the only difference is that we have to perform division instead of multiplication.

2a. Method 1: Different Values of x

When there is division between two functions and the values are different, you can first evaluate each function for the given value and then perform the division.

EXAMPLE

Find the value of fraction numerator f left parenthesis 2 right parenthesis over denominator g left parenthesis 3 right parenthesis end fraction when f open parentheses x close parentheses equals x squared minus 1 and g open parentheses x close parentheses equals 3 x.

Since the x value is different in each function, evaluate f open parentheses 2 close parentheses and g open parentheses 3 close parentheses separately, then divide the results.
f open parentheses 2 close parentheses divided by g open parentheses 3 close parentheses Evaluate f open parentheses 2 close parentheses
f open parentheses 2 close parentheses equals open parentheses 2 close parentheses squared minus 1 equals 3 Evaluate g open parentheses 3 close parentheses
g open parentheses 3 close parentheses equals 3 open parentheses 3 close parentheses equals 9 Divide f open parentheses 2 close parentheses by g open parentheses 3 close parentheses
fraction numerator f open parentheses 2 close parentheses over denominator g open parentheses 3 close parentheses end fraction equals 3 over 9 Simplify
fraction numerator f open parentheses 2 close parentheses over denominator g open parentheses 3 close parentheses end fraction equals 1 third Our solution

2b. Method 2: Same Values of x

Similar to Method 2 discussed above for multiplying two functions, if the two functions are dependent on the same variable and evaluated for the same value, we can also perform division between two functions first and then evaluate for a given value.

EXAMPLE

Find the value of fraction numerator f left parenthesis 2 right parenthesis over denominator g left parenthesis 2 right parenthesis end fraction using the same functions given above.

table attributes columnalign left end attributes row cell f open parentheses x close parentheses equals x squared minus 1 end cell row cell g open parentheses x close parentheses equals 3 x end cell end table Substitute f open parentheses x close parentheses and g open parentheses x close parentheses with their equivalent expressions
fraction numerator f open parentheses x close parentheses over denominator g open parentheses x close parentheses end fraction equals fraction numerator x squared minus 1 over denominator 3 x end fraction Substitute 2 in for x
fraction numerator f open parentheses 2 close parentheses over denominator g open parentheses 2 close parentheses end fraction equals fraction numerator 2 squared minus 1 over denominator 3 open parentheses 2 close parentheses end fraction Evaluate numerator and denominator
fraction numerator f open parentheses 2 close parentheses over denominator g open parentheses 2 close parentheses end fraction equals 3 over 6 Simplify
fraction numerator f open parentheses 2 close parentheses over denominator g open parentheses 3 close parentheses end fraction equals 1 half Our solution

As with multiplying two functions you can use either method shown above the first method works quite well for simple functions but the second method can be much more useful when working with more complex functions. Unlike with multiplication, you have to be careful that you do not divide by 0 when dividing two functions. If we have the generic form f open parentheses x close parentheses divided by g open parentheses x close parentheses comma when g open parentheses x close parentheses equals 0, the function will be undefined.

big idea
When dividing two functions, you can evaluate each function, and then divide. If both functions are defined by the same variable and evaluated at the same value, we can divide both functions first and then evaluate for the given value. In the second case, the notation used is f open parentheses a close parentheses divided by g open parentheses a close parentheses equals open parentheses f divided by g close parentheses open parentheses a close parentheses.

hint
We must be sure that the function in the denominator does not equal zero. If this does happen, then our function would be "undefined" or we would simply have "no solution." Note that the function in the denominator may return an undefined solution for only a certain domain. In our previous example, f open parentheses x close parentheses divided by g open parentheses x close parentheses would be undefined at x equals 3, because g open parentheses 3 close parentheses equals 0, however the function would have a value for other values of x.

summary
When multiplying or dividing functions, evaluate each function separately and combine the values for each function. For a given value a in the domain of f open parentheses x close parentheses and g open parentheses x close parentheses comma f open parentheses a close parentheses g open parentheses a close parentheses equals open parentheses f times g close parentheses open parentheses a close parentheses. Similarly, f open parentheses a close parentheses divided by g open parentheses a close parentheses equals open parentheses f divided by g close parentheses open parentheses a close parentheses.

Source: ADAPTED FROM "BEGINNING AND INTERMEDIATE ALGEBRA" BY TYLER WALLACE, AN OPEN SOURCE TEXTBOOK AVAILABLE AT www.wallace.ccfaculty.org/book/book.html. License: Creative Commons Attribution 3.0 Unported License

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