3.2Function Notation
Function notation f(x) is the language mathematicians use to describe and evaluate functions precisely. Learn to read, write, and evaluate functions using substitution.
Why This Matters
Function notation f(x) is the standard language of mathematics and programming. You'll see it in every Precalculus and Calculus course, and it's how functions are defined in virtually every coding language.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Essential Question
How does function notation help us communicate and evaluate mathematical relationships efficiently?
Lesson Overview
Instead of writing y = 2x + 3, mathematicians write f(x) = 2x + 3. The notation f(x) — read "f of x" — names the function f and shows that x is the input. To evaluate a function, substitute the given input value for x and simplify. Function notation makes it easy to communicate specific output values: f(4) means "find the output when x = 4." Functions can be named with any letter: f, g, h, p, etc.
Function machine: input x, rule f(x) = 2x + 3, output f(x)
Function machine: g(x) = x² − 1
Worked Examples
Reading Function Notation
f(x) = 2x + 3
f(4) means "evaluate f when x = 4"
f(4) = 2(4) + 3 = 8 + 3 = 11
Given f(x) = 3x − 5, find f(4).
Write the function rule: f(x) = 3x − 5
Substitute x = 4: f(4) = 3(4) − 5
Multiply: 3(4) = 12
Subtract: 12 − 5 = 7
Given f(x) = 3x − 5, find f(−2).
Write the function rule: f(x) = 3x − 5
Substitute x = −2 (use parentheses): f(−2) = 3(−2) − 5
Multiply: 3(−2) = −6
Subtract: −6 − 5 = −11
Given g(x) = x² + 2x − 1, find g(3).
Write the function rule: g(x) = x² + 2x − 1
Substitute x = 3: g(3) = (3)² + 2(3) − 1
Evaluate the exponent: (3)² = 9
Multiply: 2(3) = 6
Add and subtract: 9 + 6 − 1 = 14
Given h(x) = −2x² + 4, find h(−1).
Write the function rule: h(x) = −2x² + 4
Substitute x = −1: h(−1) = −2(−1)² + 4
Evaluate the exponent first: (−1)² = 1
Multiply: −2(1) = −2
Add: −2 + 4 = 2
Write a function rule for the table: x: 0, 1, 2, 3 → f(x): 4, 7, 10, 13.
Find the pattern: each output increases by 3 as x increases by 1 → slope = 3.
When x = 0, f(x) = 4 → the starting value (y-intercept) is 4.
Write the rule: f(x) = 3x + 4.
Verify: f(1) = 3(1) + 4 = 7 ✓ f(2) = 3(2) + 4 = 10 ✓
Guided Practice
Guided Practice Video: Function Notation
Watch the guided practice walkthrough for function notation, then complete the problems below.
Video by Sang Real Math
Watch on YouTube ↗Answers are in the Answer Key section.
Given f(x) = 5x + 2, find f(3).
Hint: Substitute x = 3 into the rule: f(3) = 5(3) + 2. Multiply first, then add.
Given f(x) = 5x + 2, find f(0).
Hint: Substitute x = 0: f(0) = 5(0) + 2. Anything times 0 is 0.
Given g(x) = x² − 3, find g(4).
Hint: Substitute x = 4: g(4) = (4)² − 3. Evaluate the exponent first.
Given g(x) = x² − 3, find g(−2).
Hint: Use parentheses: g(−2) = (−2)² − 3. Remember (−2)² = 4, not −4.
Given h(x) = −4x + 1, find h(−3).
Hint: Substitute x = −3: h(−3) = −4(−3) + 1. Multiplying two negatives gives a positive.
Graph of f(x) = 2x + 3 (slope 2, y-intercept 3)
Key Vocabulary
Function Notation
The notation f(x) used to name a function f with input variable x. Read as 'f of x'.
Example: f(x) = 2x + 3 means the output is 2x + 3.
Evaluate
To find the output value of a function by substituting a specific input value for x.
Example: Evaluate f(4): substitute x = 4 and simplify.
f(x)
Represents the output (y-value) of function f when the input is x. f(x) = y.
Example: f(3) = 7 means when x = 3, the output is 7.
Substitution
Replacing the variable x with a specific number to calculate the function's output.
Example: f(x) = 2x + 1; substitute x = 5: f(5) = 11.
Function Rule
An equation that describes the relationship between input and output, written in function notation.
Example: f(x) = 3x − 2 is a function rule.
Function Value
The output produced by a function for a specific input.
Example: f(3) = 7 means the function value at x = 3 is 7.
Interactive Practice — 5 Questions
Given f(x) = 5x − 2, what is f(3)?
Which statement correctly describes f(x) notation?
Given g(x) = x² − 4, what is g(−3)?
Given f(x) = 2x + 6, for what value of x does f(x) = 0?
The table shows: x: 0, 1, 2, 3 → f(x): 1, 4, 7, 10. Which function rule matches?
Independent Practice
Answers are in the Answer Key section.
Independent Practice
Given f(x) = 2x + 7, find: (a) f(1) (b) f(5) (c) f(0) (d) f(−3)
Given g(x) = −3x + 10, find: (a) g(2) (b) g(−1) (c) g(0) (d) g(4)
Given h(x) = x² + 1, find: (a) h(3) (b) h(−3) (c) h(0)
Write the function rule for the table: x: 0, 1, 2, 3 → f(x): −1, 2, 5, 8.
If f(x) = 4x − 3, what is the value of x when f(x) = 9?
Common Mistakes
Reading f(x) as 'f times x' and trying to multiply f by x.
f(x) means 'f of x' — it's the output of function f when the input is x. It's not multiplication.
Substituting the wrong value — e.g., finding f(3) but substituting x = f instead of x = 3.
f(3) means replace every x in the formula with 3. f(3) = 2(3) + 1 = 7.
Forgetting to apply order of operations after substituting — e.g., f(x) = x² + 1, f(−3) = −3² + 1 = −8 instead of 10.
(−3)² = 9, not −9. Always use parentheses when substituting negative values.
Confusing f(x + 2) with f(x) + 2 — treating them as the same expression.
f(x + 2) means substitute (x + 2) for x everywhere. f(x) + 2 means evaluate f(x) then add 2.
Math Tips
f(x) is NOT multiplication: f(x) does not mean f times x. It means "function f evaluated at x."
Substitute carefully: Replace every x in the rule with the input value, then simplify step by step.
Use parentheses: When substituting a negative number, always use parentheses: f(−2) → replace x with (−2).
f(a) = b means: When the input is a, the output is b. This is the ordered pair (a, b).
Multiple function names: g(x), h(x), p(x) all work the same way as f(x) — the letter just names the function.