1.1Understanding Functions and Their Notation
Define a function, use f(x) notation, evaluate functions algebraically and from graphs, and determine whether a relation is a function using the vertical line test.
Functions are the language of mathematics. Every formula, equation, and model you will encounter in calculus and beyond is a function — mastering notation and evaluation here unlocks everything that follows.
Essential Question
What makes a relationship between inputs and outputs a function, and how does function notation help us communicate mathematical ideas precisely?
Lesson Overview
A relation is any set of ordered pairs that pairs input values with output values. A function is a special kind of relation where each input (x-value) is paired with exactly one output (y-value). If even one input maps to two different outputs, the relation is not a function.
Function notation uses the symbol f(x), read "f of x," to represent the output of function f when the input is x. This notation is not multiplication — it is a precise way to name the output. For example, f(3) means "the value of f when x = 3."
To evaluate a function, substitute the given value in place of every x in the formula and simplify. This process is fundamental to all of calculus and applied mathematics, so mastering it now will pay dividends throughout your studies.
Key Concepts — Visual Guide
Function Notation Anatomy
How to Evaluate f(x) = 3x − 5 at x = 4
Step 1: Replace every x with 4 → 3(4) − 5
Step 2: Multiply → 12 − 5
Step 3: Subtract → f(4) = 7
Mapping Diagrams — Function vs. Non-Function
✓ Function — each input has exactly one output
✗ Not a Function — input 1 maps to two outputs
Vertical Line Test
✓ Passes — vertical line hits once
✗ Fails — vertical line hits twice
Worked Examples
Guided Practice
External Supplemental Resource
Guided Practice Video: Understanding Functions and Their Notation
Review the meaning of a function, function notation, inputs, and outputs before completing the guided practice problems.
Video by Khan Academy on YouTube
Watch on YouTube ↗Is the relation {(2,4),(3,4),(5,4)} a function?
Hint: Check if any input (x-value) repeats with a different output.
Is the relation {(1,2),(1,3),(2,4)} a function?
Hint: Does input 1 appear more than once? If so, what outputs does it map to?
Given f(x) = 4x + 7, find f(3).
Hint: Substitute x = 3 into the formula and simplify.
Given g(x) = x² + 5, find g(−2).
Hint: Remember: (−2)² = 4, not −4. Square the entire negative number.
Given h(x) = 2x² − 3x + 1, find h(0) and h(1).
Hint: Substitute each value separately. For h(0), every term with x becomes 0.
Key Vocabulary
Relation
A set of ordered pairs (x, y) that pairs input values with output values.
Function
A relation where each input (x-value) is paired with exactly one output (y-value).
Function notation
f(x) is read "f of x" and represents the output of function f when the input is x.
Domain
The set of all allowable input values (x-values) for a function.
Range
The set of all possible output values (y-values) of a function.
Vertical Line Test
A graph represents a function if and only if every vertical line crosses the graph at most once.
Check Your Understanding
Interactive Practice — 5 Questions
Which of the following relations is a function?
Given f(x) = 5x − 3, what is f(4)?
Given g(x) = x² − 4, what is g(−3)?
What does f(x) = 7 mean when x = 2?
Which statement about the vertical line test is correct?
Independent Practice
Independent Practice
Is {(3,5),(4,6),(5,7),(6,8)} a function? Explain your reasoning.
Is {(2,1),(3,1),(2,4)} a function? Explain your reasoning.
Given f(x) = 6x − 2, find f(5).
Given g(x) = x² + 3x − 4, find g(2).
Given h(x) = (x + 1)/(x − 2), find h(4).
Common Mistakes
Thinking f(x) means f times x.
f(x) is function notation — it means 'the output of f when the input is x.' It is NOT multiplication.
Evaluating (−3)² as −9.
(−3)² = (−3)(−3) = 9. The negative is inside the parentheses, so it gets squared too.
Saying a relation is not a function because two inputs give the same output.
Two different inputs CAN give the same output. A function only fails if ONE input gives TWO different outputs.
Confusing domain (inputs) with range (outputs).
Domain = x-values (inputs). Range = y-values (outputs). Think: D comes before R, just like x comes before y.
Math Tips
To evaluate f(a), replace every x in the formula with a — including inside parentheses and exponents.
When checking if a relation is a function, list all the x-values. If any x-value appears twice with different y-values, it is NOT a function.
The vertical line test is the fastest way to check a graph: if any vertical line hits the graph more than once, it fails.
SAT/ACT tip: function notation questions often ask for f(a+b) or f(f(x)) — always substitute the entire expression for x.