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

Name: How to Multiply and Divide Functions? (+FREE Worksheet!)
Uploaded: 2020-02-07T23:34:04+00:00
Description: 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 or

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.

Quick answer: To divide functions, write (f/g)(x) = f(x) ÷ g(x), simplify if you can, and exclude every x that makes g(x) = 0. To multiply, use (f · g)(x) = f(x) · g(x). Example: if f(x) = x² − 9 and g(x) = x − 3, then (f/g)(x) = (x + 3)(x − 3)/(x − 3) = x + 3, with x ≠ 3.

Tutor-style math help

Multiplying and Dividing Functions: what to notice and how to work it

Functions skill

A function is a rule that gives each input exactly one output. Function notation, tables, graphs, and equations are different ways to show the same input-output relationship.

What to notice first

Ask what kind of input you are given. Sometimes you substitute a number, sometimes you read a graph, and sometimes you combine two rules.

Common student mistake

Do not read \(f(4)\) as multiplication. It means the output of f when the input is 4.

Key formulas and cues

\(f(a)\text{ means replace }x\text{ with }a\)

\((f\circ g)(x)=f(g(x))\)

\(f^{-1}(x)\text{ reverses }f(x)\)

A reliable path

Identify the inputFind the x-value, expression, or inner function being used.
Apply the ruleSubstitute with parentheses so signs and powers stay clear.
Interpret the outputState the value, point, interval, domain, range, or inverse relationship.

Worked examples

Evaluate a function

Example: \(f(x)=4x-3\), find \(f(2)\)

Replace x with 2.
Compute 4(2) – 3.
Simplify.

Answer: \(5\)

Compose functions

Example: \(f(x)=x+1\), \(g(x)=2x\), find \(f(g(3))\)

Find g(3) = 6.
Use that as the input for f.
f(6) = 7.

Answer: \(7\)

Open pop-up practice

Try one before moving on

Try: If \(h(x)=2x^2\), find \(h(-3)\).

Answer: \(18\). Use parentheses: \(2(-3)^2=18\).

Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.

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

Write both function rules.
To multiply: multiply the expressions and expand using FOIL or distribution.
To divide: write f(x) over g(x) and factor both to cancel common factors.
State any domain restrictions (values where \(\color{blue}{g(x) = 0}\) are excluded).
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.

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

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Answers

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

How to Find the Domain When Dividing Functions

The domain of (f/g)(x) is every x that is in the domain of f and in the domain of g, minus the values that make g(x) equal 0. Find those values before you simplify. Canceling a factor does not bring a forbidden x back.

Write the quotient f(x)/g(x).
Set g(x) = 0 and solve. These x-values are excluded.
Factor the top and bottom and cancel any common factors.
State the simplified rule together with the excluded values.

Example 1: f(x) = x + 1 and g(x) = x² − 9. Solving x² − 9 = 0 gives x = 3 or x = −3. So (f/g)(x) = (x + 1)/(x² − 9), with x ≠ 3 and x ≠ −3.

Example 2: f(x) = x² + 5x + 6 and g(x) = x + 3. First, g(x) = 0 when x = −3. Factor the top: x² + 5x + 6 = (x + 2)(x + 3). Cancel (x + 3) to get (f/g)(x) = x + 2, with x ≠ −3. The graph is the line y = x + 2 with a hole at (−3, −1).

Two ways to evaluate a quotient at a number

To find (f/g)(2) in Example 2, you can plug in first: f(2) = 4 + 10 + 6 = 20 and g(2) = 5, so 20 ÷ 5 = 4. Or use the simplified rule: 2 + 2 = 4. Both give the same answer. If the number is an excluded value, such as −3, the answer is undefined.

Frequently Asked Questions

How do you multiply and divide functions?

To multiply, use (f · g)(x) = f(x) · g(x); to divide, use (f/g)(x) = f(x) ÷ g(x) and exclude any x that makes g(x) = 0. With f(x) = x + 1 and g(x) = x², (f · g)(x) = x³ + x². Check at x = 2: f(2) · g(2) = 3 · 4 = 12, and 2³ + 2² = 12 too.

How do you divide two functions?

Write f(x) over g(x), factor both if you can, and cancel common factors. Then state that x cannot be any value that makes g(x) = 0, even if that factor cancels. For f(x) = x² − 4 and g(x) = x + 2, (f/g)(x) = (x − 2)(x + 2)/(x + 2) = x − 2, with x ≠ −2.

How do you multiply and divide rational functions?

Factor every numerator and denominator, multiply straight across (or flip the second fraction to divide), then cancel common factors. For example, (x² − 1)/(x + 3) · (x + 3)/(x − 1) = (x − 1)(x + 1)(x + 3)/((x + 3)(x − 1)) = x + 1. Keep the restrictions from the original denominators: x ≠ −3 and x ≠ 1.

How do you add, subtract, multiply and divide functions?

Do the operation on the two rules for the same x. With f(x) = 2x and g(x) = x − 5: (f + g)(x) = 3x − 5, (f − g)(x) = x + 5, (f · g)(x) = 2x² − 10x and (f/g)(x) = 2x/(x − 5). Only division adds a new restriction, here x ≠ 5. Use parentheses when subtracting so signs stay correct.

How do you multiply and divide polynomials?

To multiply, distribute every term of one polynomial to every term of the other and combine like terms: (x + 3)(x − 2) = x² + x − 6. To divide by a monomial, divide each term separately: (6x³ − 9x²) ÷ 3x = 2x² − 3x, with x ≠ 0. For a binomial divisor, factor or use long division.

Is there a multiplying and dividing functions worksheet?

Yes. A good worksheet gives pairs of functions and asks for (f · g)(x), (f/g)(x) and their values at specific numbers, such as (f/g)(4). Make sure it also asks for domain restrictions, since that is the step students skip most often. Check one or two answers by plugging in a number for x.

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