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Adding and Subtracting Functions

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

Table of Contents

1. Adding Two Functions

Suppose we were given two functions f open parentheses x close parentheses and g open parentheses x close parentheses and asked to add them together. Because both of these functions are given in terms of the variable x (remember that f open parentheses x close parentheses means "x is an argument of the function f") we can combine the functions together by adding the like terms in each.

EXAMPLE

Suppose f open parentheses x close parentheses equals 2 x squared minus x plus 5 and g open parentheses x close parentheses equals x squared minus 5 x plus 1. If we wanted to find f open parentheses x close parentheses plus g open parentheses x close parentheses, we simply do the following:

f open parentheses x close parentheses plus g open parentheses x close parentheses equals open parentheses 2 x squared minus x plus 5 close parentheses plus open parentheses x squared minus 5 x plus 1 close parentheses Group like terms
f open parentheses x close parentheses plus g open parentheses x close parentheses equals 2 x squared plus x squared minus x minus 5 x plus 5 plus 1 Combine like terms
f open parentheses x close parentheses plus g open parentheses x close parentheses equals 3 x squared minus 6 x plus 6 Our solution

hint
When trying to add two functions defined by the same variable often times you will see the notation open parentheses f plus g close parentheses open parentheses x close parentheses, which just means f open parentheses x close parentheses plus g open parentheses x close parentheses.

Sometimes we may be asked to evaluate two functions for different values first and then add the result.

EXAMPLE

Suppose f open parentheses x close parentheses equals 2 x minus 1 and g open parentheses x close parentheses equals 3 x. Find the result of f open parentheses 2 close parentheses plus g open parentheses 1 close parentheses. In cases like this, since the value of x is different, we first need to evaluate each function of the given value and then add the final results, as shown below:

f open parentheses 2 close parentheses plus g open parentheses 1 close parentheses First evaluate f open parentheses 2 close parentheses
f open parentheses 2 close parentheses equals 2 open parentheses 2 close parentheses minus 1 equals 3 Now evaluate g open parentheses 1 close parentheses
g open parentheses 1 close parentheses equals 3 open parentheses 1 close parentheses equals 3 Add f open parentheses 2 close parentheses and g open parentheses 1 close parentheses together
f open parentheses 2 close parentheses plus g open parentheses 1 close parentheses equals 3 plus 3 Simplify
f open parentheses 2 close parentheses plus g open parentheses 1 close parentheses equals 6 Our solution

2. Subtracting Two Functions

When subtracting two functions we follow the same rules outlined above for addition only this time we have a negative sign between the two functions.

EXAMPLE

Suppose we had the functions f open parentheses x close parentheses equals x minus 1 and g open parentheses x close parentheses equals x cubed minus 2 x plus 1 and asked to find f open parentheses x close parentheses minus g open parentheses x close parentheses we would need to do the following:

f open parentheses x close parentheses minus g open parentheses x close parentheses equals open parentheses x minus 1 close parentheses minus open parentheses x cubed minus 2 x plus 1 close parentheses Subtract functions by distributing the negative
f open parentheses x close parentheses minus g open parentheses x close parentheses equals x minus 1 minus x cubed plus 2 x minus 1 Group like terms
f open parentheses x close parentheses minus g open parentheses x close parentheses equals short dash x cubed plus x plus 2 x minus 1 minus 1 Combine like terms
f open parentheses x close parentheses minus g open parentheses x close parentheses equals short dash x cubed plus 3 x minus 2 Our solution

hint
Always remember to distribute the negative when subtracting functions, otherwise your calculation will be incorrect.

EXAMPLE

Find f open parentheses 3 close parentheses minus g open parentheses 3 close parentheses for the functions f open parentheses x close parentheses equals x minus 1 and g open parentheses x close parentheses equals x cubed minus 2 x plus 1.

We can use the same method we did above since both functions are being evaluated for the same value of x and both are in terms of x. We can use the solution in the last step of the above example, f open parentheses x close parentheses minus g open parentheses x close parentheses equals short dash x cubed plus 3 x minus 2, and simply substitute 3 for x and solve evaluate.

f open parentheses x close parentheses minus g open parentheses x close parentheses equals short dash x cubed plus 3 x minus 2 Substitute 3 in for x
f open parentheses 3 close parentheses minus g open parentheses 3 close parentheses equals short dash open parentheses 3 cubed close parentheses plus 3 open parentheses 3 close parentheses minus 2 Evaluate terms
f open parentheses 3 close parentheses minus g open parentheses 3 close parentheses equals short dash 27 plus 9 minus 2 Simplify
f open parentheses 3 close parentheses minus g open parentheses 3 close parentheses equals short dash 20 Our solution

We could also evaluate each function of the given value and then subtract the final results, as shown below:

table attributes columnalign left end attributes row cell f open parentheses x close parentheses equals x minus 1 end cell row cell g open parentheses x close parentheses equals x cubed minus 2 x plus 1 end cell end table First evaluate f open parentheses 3 close parentheses
f open parentheses 3 close parentheses equals 3 minus 1 equals 2 Then evaluate g open parentheses 3 close parentheses
g open parentheses 3 close parentheses equals 3 cubed minus 2 open parentheses 3 close parentheses plus 1 equals 27 minus 6 plus 1 equals 22 Subtract f open parentheses 3 close parentheses and g open parentheses 3 close parentheses
f open parentheses 3 close parentheses minus g open parentheses 3 close parentheses equals 2 minus 22 Simplify
f open parentheses 3 close parentheses minus g open parentheses 3 close parentheses equals short dash 20 Our solution

hint
When trying to subtract two functions defined by the same variable often times you will see the notation open parentheses f minus g close parentheses open parentheses x close parentheses, which just means f open parentheses x close parentheses minus g open parentheses x close parentheses.

Keep in mind that if f open parentheses x close parentheses is defined using different variables or for different values and you are asked to find f open parentheses x close parentheses minus g open parentheses x close parentheses, you first need to evaluate each function for the given value of its variable separately and then subtract the final results.

summary
When adding or subtracting two 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 plus g open parentheses a close parentheses equals open parentheses f plus g close parentheses open parentheses a close parentheses and f open parentheses a close parentheses minus g open parentheses a close parentheses equals open parentheses f minus 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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