How to Add and Subtract Functions? (+FREE Worksheet!)

When you know how to add and subtract numbers, you can apply those same ideas to adding and subtracting functions. Adding and subtracting functions means combining two function rules into one new function, and all you need to do is add or subtract the formulas and then simplify. This skill is essential for Algebra 1 and for every higher-level math course that builds on it.

What Does It Mean to Add or Subtract Functions?

If f and g are two functions, you can form two new functions:

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• Sum: \(\color{blue}{(f + g)(x) = f(x) + g(x)}\)
• Difference: \(\color{blue}{(f – g)(x) = f(x) – g(x)}\)

The domain of the new function is the set of all x-values that are in both the domain of f and the domain of g (the intersection). For polynomials, the domain is always all real numbers.

How to Add and Subtract Functions

Rule 1 — Add the function rules

Write \(\color{blue}{f(x) + g(x)}\), then combine like terms.

Example: \(\color{blue}{f(x) = 3x + 2}\) and \(\color{blue}{g(x) = x}\)\(\color{blue}{^{2} – x}\).
\(\color{blue}{(f + g)(x) = (3x + 2) + (x^{2} – x) = x^{2} + 2x + 2}\)

Rule 2 — Subtract the function rules

Write \(\color{blue}{f(x) – g(x)}\), distribute the minus sign through the second rule, then combine like terms.

Example: \(\color{blue}{f(x) = 3x + 2}\) and \(\color{blue}{g(x) = x}\)\(\color{blue}{^{2} – x}\).
\(\color{blue}{(f – g)(x) = (3x + 2) – (x^{2} – x) = 3x + 2 – x^{2} + x = -x^{2} + 4x + 2}\)

Rule 3 — Evaluate at a specific input

Either find the new combined function first and then substitute, or evaluate each function separately and then add or subtract.

Example: Using f and g above, find (\(\color{blue}{f + g}\))(2):
\(\color{blue}{f(2) = 3(2) + 2 = 8}\)
\(\color{blue}{g(2) = (2)^{2} – 2 = 2}\)
\(\color{blue}{(f + g)(2) = 8 + 2 = 10}\)

Step-by-Step Summary

1. Write out f(x) and g(x) side by side.
2. For addition, place a + between them; for subtraction, place a −.
3. Distribute the minus sign if subtracting (change every sign in g(x)).
4. Combine like terms and write the simplified result.
5. If asked for a specific value, substitute the input into the simplified function.

Watch: Adding and Subtracting Functions (Video Lesson)

This clear lesson covers the rules for adding and subtracting functions with worked examples:

Adding and Subtracting Functions – Worked Examples

Example 1: Given \(\color{blue}{f(x) = 3x + 2}\) and \(\color{blue}{g(x) = x}\)\(\color{blue}{^{2} – x}\), find (\(\color{blue}{f + g}\))(x).

\(\color{blue}{(f + g)(x) = (3x + 2) + (x^{2} – x)}\) = x² + \(\color{blue}{2x + 2}\)

Example 2: Given \(\color{blue}{f(x) = 3x + 2}\) and \(\color{blue}{g(x) = x}\)\(\color{blue}{^{2} – x}\), find (\(\color{blue}{f – g}\))(3).

First find (\(\color{blue}{f – g}\))(x):
\(\color{blue}{(f – g)(x) = (3x + 2) – (x^{2} – x)}\) = −x² + \(\color{blue}{4x + 2}\)
Substitute \(\color{blue}{x = 3}\):
\(\color{blue}{(f – g)(3) = -(3)^{2} + 4(3) + 2 = -9 + 12 + 2 = 5}\)
(Check: \(\color{blue}{f(3) = 11}\), \(\color{blue}{g(3) = 6}\), \(\color{blue}{11 – 6 = 5}\) ✓)

Example 3: Given \(\color{blue}{f(x) = 2x + 1}\) and \(\color{blue}{g(x) = x – 3}\), find (\(\color{blue}{f + g}\))(4).

\(\color{blue}{f(4) = 2(4) + 1 = 9}\);  \(\color{blue}{g(4) = 4 – 3 = 1}\)
\(\color{blue}{(f + g)(4) = 9 + 1 = 10}\)

Example 4: Given \(\color{blue}{f(x) = 2x + 1}\) and \(\color{blue}{g(x) = x – 3}\), find (\(\color{blue}{f – g}\))(4).

\(\color{blue}{f(4) = 9}\);  \(\color{blue}{g(4) = 1}\)
\(\color{blue}{(f – g)(4) = 9 – 1 = 8}\)

More Practice: Step-by-Step Video Review

Brian McLogan walks through adding, subtracting, multiplying, and dividing functions with multiple examples:

Exercises for Adding and Subtracting Functions

Let \(\color{blue}{f(x) = 3x + 2}\) and \(\color{blue}{g(x) = x}\)\(\color{blue}{^{2} – x}\). Find each of the following.

1. (\(\color{blue}{f + g}\))(x) in simplified form
2. (\(\color{blue}{f – g}\))(x) in simplified form
3. (\(\color{blue}{f + g}\))(1)
4. (\(\color{blue}{f – g}\))(4)
5. (\(\color{blue}{g – f}\))(0)

Answers

1. (\(\color{blue}{f + g}\))(x) = x² + \(\color{blue}{2x + 2}\)
2. (\(\color{blue}{f – g}\))(x) = −x² + \(\color{blue}{4x + 2}\)
3. \(\color{blue}{f(1) = 5}\), \(\color{blue}{g(1) = 0}\), so (\(\color{blue}{f + g}\))(1) = 5
4. \(\color{blue}{f(4) = 14}\), \(\color{blue}{g(4) = 12}\), so \(\color{blue}{(f – g)(4) = 14 – 12}\) = 2
5. \(\color{blue}{g(0) = 0}\), \(\color{blue}{f(0) = 2}\), so \(\color{blue}{(g – f)(0) = 0 – 2}\) = −2

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Free Adding and Subtracting Functions Worksheet

Ready to practice on your own? Download our free Adding and Subtracting Functions worksheet below, work through each problem at your own pace, and then check your answers. If a few give you trouble, scroll back up to the worked examples and try again — steady practice is the surest way to master Adding and Subtracting Functions before a quiz or test.

Download Combining Functions Worksheet

Frequently Asked Questions

Is (\(\color{blue}{f + g}\))(x) the same as f(g(x))?

No. \(\color{blue}{(f + g)(x) = f(x) + g(x)}\) is addition of two functions. f(g(x)) is composition (plugging g into f), which is a completely different operation.

Do I need to distribute the minus sign when subtracting functions?

Yes. When computing (\(\color{blue}{f – g}\))(x), you must subtract the entire expression g(x), which means changing the sign of every term in g(x). Forgetting to distribute is the most common mistake.

What is the domain of (\(\color{blue}{f + g}\))?

The domain of (\(\color{blue}{f + g}\)) is the intersection of the domains of f and g — in other words, all x-values that are valid inputs for both functions at the same time. For polynomial functions, this is always all real numbers.

Related Topics

• Function Notation
• Multiplying and Dividing Functions
• Composition of Functions
• Function Inverses
• Finding Domain and Range of a Function