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

How do you add and subtract functions?

Add or subtract the two function rules, then combine like terms. If f(x) = 3x + 2 and g(x) = x − 5, then (f + g)(x) = 3x + 2 + x − 5 = 4x − 3. For subtraction, put g(x) in parentheses first: (f − g)(x) = 3x + 2 − (x − 5) = 2x + 7. Forgetting to subtract every term of g(x) is the most common slip.

How do you add and subtract rational functions?

Treat them like fractions: find a common denominator, rewrite each function over it, then add or subtract the numerators. For example, 1/x + 2/(x + 1) becomes (x + 1)/(x(x + 1)) + 2x/(x(x + 1)) = (3x + 1)/(x(x + 1)). Remember that the new function is undefined wherever either original denominator is zero, here x = 0 and x = −1.

How do you add, subtract and multiply functions?

Each operation works on the outputs. With f(x) = x + 2 and g(x) = x − 3, you get (f + g)(x) = 2x − 1 and (f − g)(x) = x + 2 − x + 3 = 5. For multiplication, (f · g)(x) = (x + 2)(x − 3) = x² − x − 6. Simplify each result and write it in standard form, highest power first.

How do you add and subtract functions graphically?

Work one x-value at a time: read f(x) and g(x) from their graphs, then add or subtract those y-values. If f(2) = 3 and g(2) = 1, the point (2, 4) lies on the graph of f + g and (2, 2) lies on the graph of f − g. Repeat for several x-values and connect the new points to sketch the combined function.

How do you add and subtract trig functions?

Combine them the same way you combine like terms in algebra. For f(x) = 3 sin x and g(x) = 5 sin x, (f + g)(x) = 8 sin x and (f − g)(x) = −2 sin x. Unlike terms such as sin x and cos x stay separate, so sin x + 2 cos x is already simplified unless you use an identity to rewrite it.

Is there an adding and subtracting functions worksheet with answers?

Yes. The free adding and subtracting functions worksheet comes with an answer key so students can check each step. A good habit is to rewrite every problem in parentheses before simplifying, for example (4x² − x) − (x² + 3x) = 3x² − 4x, and then compare with the key to spot sign mistakes early.

Related Topics

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