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 is function notation in math?

Function notation is a way of naming a function and its input, written f(x) and read “f of x.” It means the output of function f when the input is x. Writing f(x) = 2x + 3 says the same thing as y = 2x + 3, but it also lets you ask for specific outputs, like f(5) = 13.

How do you evaluate function notation?

Replace every x in the rule with the given input, using parentheses, then simplify. For f(x) = 2x + 3, f(4) = 2(4) + 3 = 11 and f(−1) = 2(−1) + 3 = 1. Parentheses matter most with squares: if g(x) = x² + 1, then g(−3) = (−3)² + 1 = 10, not −8.

How do you find function notation from a graph?

To find f(a), go to x = a on the graph and read the y-value there. If the graph passes through (3, 5), then f(3) = 5. To solve f(x) = 2 instead, look for every point where the graph’s height is 2 and read the matching x-values; there may be more than one.

Is f(x) the same as y in function notation?

In most Algebra 1 problems, yes: f(x) plays the role of y, the output. The difference is that f(x) also names the function and shows the input. Writing y = 7 says nothing about the input, but f(2) = 7 tells you that an input of 2 produces an output of 7. Also, f(x) never means f times x.

Why is function notation important?

It keeps track of which function and which input you are using, which matters once a problem has more than one rule. With f(x) = x + 4 and g(x) = 3x, you can quickly compare f(2) = 6 with g(2) = 6, or build new functions like f(g(x)). That clarity is why function notation appears in Algebra 1, Algebra 2, and calculus.

How do you write function notation?

Start with an equation solved for y and replace y with a function name such as f(x). For example, y = x² − 4 becomes f(x) = x² − 4. From a word problem, name the input clearly: if a taxi charges $3 plus $2 per mile, the cost function is C(m) = 2m + 3, so C(5) = $13.

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