How to Solve Function Notation? (+FREE Worksheet!)

Function notation is the standard language mathematicians use to describe and evaluate functions. Instead of writing “y equals 2x plus 3,” we write \(\color{blue}{f(x) = 2x + 3}\), which makes it clear we are talking about a function named f that depends on the variable x. Once you understand function notation, evaluating functions becomes a simple substitution exercise that appears throughout Algebra 1 and beyond.

What Is Function Notation?

Function notation uses the form f(x), read “f of x,” to show that the output of the function depends on the input x. The letter f is the name of the function (other common names are g, h, or any letter). The expression inside the parentheses is the input (also called the argument), and the rule after the equals sign tells you how to calculate the output.

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• \(\color{blue}{f(x) = 2x + 3}\) means “take the input, multiply by 2, then add 3.”
• f(4) means “evaluate the function when \(\color{blue}{x = 4}\).”

How to Evaluate a Function Using Function Notation

Step 1 — Identify the input value

Read what is inside the parentheses. If the problem asks for f(4), the input is 4.

Step 2 — Substitute the input into the rule

Replace every occurrence of the variable in the formula with the given input.

Example: Given \(\color{blue}{f(x) = 2x + 3}\), evaluate f(4):
\(\color{blue}{f(4) = 2(4) + 3 = 8 + 3 = 11}\)

Step 3 — Simplify

Carry out the arithmetic in the correct order of operations.

Example: Given \(\color{blue}{g(x) = x}\)\(\color{blue}{^{2} – 1}\), evaluate g(5):
\(\color{blue}{g(5) = 5^{2} – 1 = 25 – 1 = 24}\)

Evaluating with negative inputs

Always place negative inputs in parentheses to avoid sign errors.

Example: Given \(\color{blue}{f(x) = 2x + 3}\), evaluate f(−3):
\(\color{blue}{f(-3) = 2(-3) + 3 = -6 + 3 = -3}\)

Step-by-Step Summary

1. Write the function rule.
2. Replace the variable with the given input value (use parentheses for negatives).
3. Simplify using order of operations.
4. State the output: for example, \(\color{blue}{f(4) = 11}\).

Watch: Evaluating with Function Notation (Video Lesson)

This Khan Academy video explains what function notation means and demonstrates how to evaluate functions step by step:

Function Notation – Worked Examples

Example 1: Let \(\color{blue}{f(x) = 2x + 3}\). Find f(4).

Substitute \(\color{blue}{x = 4}\):
\(\color{blue}{f(4) = 2(4) + 3 = 8 + 3 = 11}\)

Example 2: Let \(\color{blue}{g(x) = x}\)\(\color{blue}{^{2} – 1}\). Find g(−2).

Substitute x = −2:
\(\color{blue}{g(-2) = (-2)^{2} – 1 = 4 – 1 = 3}\)

Example 3: Let \(\color{blue}{f(x) = 2x + 3}\). Find f(0).

Substitute \(\color{blue}{x = 0}\):
\(\color{blue}{f(0) = 2(0) + 3 = 0 + 3 = 3}\)

Example 4: Let \(\color{blue}{g(x) = x}\)\(\color{blue}{^{2} – 1}\). Find g(5).

Substitute \(\color{blue}{x = 5}\):
\(\color{blue}{g(5) = (5)^{2} – 1 = 25 – 1 = 24}\)

More Practice: What Are Functions? (Video Lesson)

Math Antics breaks down the concept of functions and function notation with clear visual examples:

Exercises for Function Notation

Let \(\color{blue}{f(x) = 2x + 3}\) and \(\color{blue}{g(x) = x}\)\(\color{blue}{^{2} – 1}\). Evaluate each expression.

1. f(7)
2. g(0)
3. f(−5)
4. g(3)
5. f(½)

Answers

1. \(\color{blue}{f(7) = 2(7) + 3}\) = 17
2. \(\color{blue}{g(0) = 0}\)\(\color{blue}{^{2} – 1}\) = −1
3. \(\color{blue}{f(-5) = 2(-5) + 3}\) = −\(\color{blue}{10 + 3}\) = −7
4. \(\color{blue}{g(3) = 3}\)² − \(\color{blue}{1 = 9 – 1}\) = 8
5. \(\color{blue}{f(\frac{1}{2}) = 2(\frac{1}{2}) + 3 = 1 + 3}\) = 4

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Frequently Asked Questions

What does f(x) mean?

f(x) is read “f of x.” It means the output of function f when the input is x. It does not mean f times x.

Can a function be named something other than f?

Yes. Functions are commonly named g, h, p, or any letter. The name just identifies which function you are working with when several functions appear in the same problem.

What if the input itself is an expression, like f(\(\color{blue}{x + 1}\))?

Replace every x in the formula with the entire expression (\(\color{blue}{x + 1}\)), placing it in parentheses. For \(\color{blue}{f(x) = 2x + 3}\), \(\color{blue}{f(x + 1) = 2(x + 1) + 3 = 2x + 5}\).

Related Topics

• Adding and Subtracting Functions
• Multiplying and Dividing Functions
• Composition of Functions
• Function Inverses
• Finding Domain and Range of a Function
• Average Rate of Change of a Function