How to Multiply and Dividing Functions? (+FREE Worksheet!)

Once you are comfortable adding and subtracting functions, the next step is multiplying and dividing functions. Multiplying and dividing functions combines two function rules using multiplication or division, producing a new function whose output at any input is the product or quotient of the two original outputs. These operations show up frequently in Algebra 1 when working with polynomials and rational expressions.

What Are Multiplied and Divided Functions?

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

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• Product: \(\color{blue}{(f \cdot g)(x) = f(x)}\) · g(x)
• Quotient: \(\color{blue}{(\frac{f}{g})(x) = \frac{f(x)}{g(x)}}\), where g(x) ≠ 0

For the quotient, you must exclude any x-value that makes \(\color{blue}{g(x) = 0}\) from the domain, because division by zero is undefined.

How to Multiply and Divide Functions

Multiplying functions

Write the two expressions side by side and multiply, then expand and simplify.

Example: \(\color{blue}{f(x) = x + 2}\), \(\color{blue}{g(x) = x – 3}\).
\(\color{blue}{(f \cdot g)(x) = (x + 2)(x – 3) = x^{2} – x – 6}\)

To evaluate at a specific input, substitute after multiplying (or multiply the two outputs directly):
\(\color{blue}{(f \cdot g)(5) = (5 + 2)(5 – 3) = 7 \cdot 2 = 14}\)

Dividing functions

Write f(x) in the numerator and g(x) in the denominator. Simplify by cancelling common factors, and note any domain restrictions.

Example: \(\color{blue}{f(x) = x + 2}\), \(\color{blue}{g(x) = x – 3}\).
\(\color{blue}{(\frac{f}{g})(x) = \frac{(x + 2)}{(x – 3)}, x \ne 3}\)
To evaluate at \(\color{blue}{x = 4}\):
\(\color{blue}{(\frac{f}{g})(4) = \frac{(4 + 2)}{(4 – 3)} = \frac{6}{1} = 6}\)

Simplifying the quotient

Always factor numerator and denominator to cancel any shared factors.

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

Step-by-Step Summary

1. Write both function rules.
2. To multiply: multiply the expressions and expand using FOIL or distribution.
3. To divide: write f(x) over g(x) and factor both to cancel common factors.
4. State any domain restrictions (values where \(\color{blue}{g(x) = 0}\) are excluded).
5. To evaluate at a specific x, substitute into the simplified expression.

Watch: Multiplying and Dividing Functions (Video Lesson)

Brian McLogan demonstrates all four function operations, including multiplication and division, with clear examples:

Multiplying and Dividing Functions – Worked Examples

Example 1: Let \(\color{blue}{f(x) = x + 2}\) and \(\color{blue}{g(x) = x – 3}\). Find (f · g)(x).

Multiply: \(\color{blue}{(x + 2)(x – 3) = x}\)² − \(\color{blue}{3x + 2x – 6}\) = x² − \(\color{blue}{x – 6}\)

Example 2: Using the same f and g, find (f · g)(0).

\(\color{blue}{(f \cdot g)(0) = (0 + 2)(0 – 3) = 2 \cdot (-3) = -6}\)

Example 3: Using the same f and g, find (\(\color{blue}{\frac{f}{g}}\))(4).

\(\color{blue}{(\frac{f}{g})(4) = \frac{(4 + 2)}{(4 – 3)} = \frac{6}{1} = 6}\)

Example 4: Using the same f and g, evaluate (\(\color{blue}{\frac{f}{g}}\))(1). Note the domain restriction.

\(\color{blue}{(\frac{f}{g})(1) = \frac{(1 + 2)}{(1 – 3)} = \frac{3}{(-2)} = -\frac{3}{2}}\)
(Domain restriction: x ≠ 3.)

More Practice: Rational Expressions Video Review

Mario’s Math Tutoring shows how to multiply and divide rational expressions, the natural next step after multiplying and dividing function rules:

Exercises for Multiplying and Dividing Functions

Let \(\color{blue}{f(x) = x + 2}\) and \(\color{blue}{g(x) = x – 3}\). Evaluate each expression.

1. (f · g)(x) in simplified form
2. (f · g)(4)
3. (f · g)(−1)
4. (\(\color{blue}{\frac{f}{g}}\))(5) (state the domain restriction)
5. (\(\color{blue}{\frac{f}{g}}\))(0) (state the domain restriction)

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Answers

1. (f · g)(x) = x² − \(\color{blue}{x – 6}\)
2. \(\color{blue}{(4 + 2)(4 – 3) = 6}\) · 1 = 6
3. \(\color{blue}{(-1 + 2)(-1 – 3) = 1}\) · (−4) = −4
4. \(\color{blue}{\frac{(5 + 2)}{(5 – 3)} = \frac{7}{2}}\) = \(\color{blue}{\frac{7}{2}}\); domain restriction x ≠ 3
5. \(\color{blue}{\frac{(0 + 2)}{(0 – 3)} = \frac{2}{(-3)}}\) = −\(\color{blue}{\frac{2}{3}}\); domain restriction x ≠ 3

Free Multiplying and Dividing Functions Worksheet

Ready to practice on your own? Download our free Multiplying and Dividing 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 Multiplying and Dividing Functions before a quiz or test.

Download Combining Functions Worksheet

Frequently Asked Questions

Is (f · g)(x) the same as f(g(x))?

No. \(\color{blue}{(f \cdot g)(x) = f(x) \times g(x)}\) is multiplication of outputs. f(g(x)) is function composition — plugging g into f — which gives a different result.

Why do I need domain restrictions when dividing functions?

Division by zero is undefined. If \(\color{blue}{g(x) = 0}\) for some value of x, that x-value must be excluded from the domain of (\(\color{blue}{\frac{f}{g}}\)). Always solve \(\color{blue}{g(x) = 0}\) to find these restrictions.

How do I simplify (\(\color{blue}{\frac{f}{g}}\))(x) when both are polynomials?

Factor both the numerator f(x) and the denominator g(x) completely, cancel any common factors, and record the domain restriction for every cancelled factor.

Related Topics

• Function Notation
• Adding and Subtracting Functions
• Composition of Functions
• Multiplying Rational Expressions
• Dividing Rational Expressions