3.1Introduction to Functions
A function is a special relationship where every input has exactly one output. Learn to identify functions from tables, mapping diagrams, and graphs using the vertical line test.
Why This Matters
Functions are the central idea of all higher mathematics. Every graph in Precalculus, every model in Physics, and every algorithm in computer science is built on the concept of a function — this lesson is where it all begins.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Essential Question
What makes a relationship between inputs and outputs a function?
Lesson Overview
A relation is any set of ordered pairs (x, y) that pairs input values with output values. A function is a special type of relation where each input (x-value) is paired with exactly one output (y-value). The same output can appear more than once, but the same input cannot produce two different outputs. We can represent functions as tables, mapping diagrams, sets of ordered pairs, or graphs.
Each input → exactly one output
Input 2 maps to two outputs — NOT a function
Worked Examples
Parabola — passes vertical line test (function)
Circle — fails vertical line test (not a function)
Is this a function? {(1, 3), (2, 5), (3, 7), (4, 9)}
List all x-values (inputs): 1, 2, 3, 4
Check: does any x-value repeat? No — each input appears exactly once.
Each input has exactly one output → this IS a function.
Is this a function? {(2, 4), (3, 7), (2, 9), (5, 1)}
List all x-values (inputs): 2, 3, 2, 5
Check: does any x-value repeat? Yes — x = 2 appears twice (paired with 4 and with 9).
One input maps to two different outputs → this is NOT a function.
Is this table a function? x: 1, 2, 3, 4 | y: 5, 5, 8, 10
Check the x-column: 1, 2, 3, 4 — no repeats.
Note: x = 1 and x = 2 both map to y = 5. That is allowed.
Each input has exactly one output → this IS a function.
Apply the Vertical Line Test: A circle centered at the origin.
Draw a vertical line through the center of the circle.
The vertical line crosses the circle at two points (top and bottom).
Since one vertical line hits the graph twice, it fails the test.
Apply the Vertical Line Test: The line y = 2x + 1.
Draw any vertical line across the graph of y = 2x + 1.
A straight non-vertical line can only be crossed once by any vertical line.
Every vertical line hits the graph exactly once → passes the test.
Guided Practice
Answers are in the Answer Key section.
Guided Practice Video: Introduction to Functions
Watch the guided practice walkthrough for introduction to functions, then complete the problems below.
Video by Sang Real Math
Watch on YouTube ↗Is {(0, 1), (1, 3), (2, 5), (3, 7)} a function? Explain.
Hint: List the x-values. Do any repeat? If no x-value repeats, it is a function.
Is {(4, 2), (4, −2), (9, 3), (9, −3)} a function? Explain.
Hint: Check the x-values: 4 appears twice. Does it map to the same output both times?
Table: x = {1, 2, 3, 4}, y = {6, 7, 6, 8}. Is this a function?
Hint: Check the x-column for repeats. Then check: are any x-values paired with two different y-values?
Table: x = {5, 6, 5, 7}, y = {2, 4, 9, 1}. Is this a function?
Hint: x = 5 appears twice. What are its y-values? If they differ, it is NOT a function.
A mapping diagram shows: 1→3, 2→5, 3→5, 4→7. Is this a function?
Hint: Each input (1, 2, 3, 4) has exactly one arrow. Does any input split into two outputs?
Plotting ordered pairs: {(1,3), (2,5), (3,7), (4,9)}
Key Vocabulary
Relation
Any set of ordered pairs (x, y) that pairs input values with output values.
Example: {(1,2),(3,4)} is a relation.
Function
A relation where each input (x-value) is paired with exactly one output (y-value).
Example: {(1,3),(2,5),(3,7)} is a function.
Input / Domain
The x-values (independent variable) put into a function.
Example: In {(1,3),(2,5)}, the domain is {1,2}.
Output / Range
The y-values (dependent variable) that result from a function.
Example: In {(1,3),(2,5)}, the range is {3,5}.
Mapping Diagram
A diagram using arrows to show how each input is paired with an output.
Example: 1→3, 2→5 shows each input has one output.
Vertical Line Test
If any vertical line crosses a graph more than once, the graph is NOT a function.
Example: A circle fails; y = 2x + 1 passes.
Interactive Practice — 5 Questions
Which of the following is the definition of a function?
Which set of ordered pairs is NOT a function?
A mapping diagram shows: 3→7, 4→9, 5→7, 6→11. Is this a function?
Which statement about the vertical line test is correct?
Which real-world situation represents a function?
Independent Practice
Answers are in the Answer Key section.
Independent Practice
Is {(2, 5), (4, 9), (6, 13), (8, 17)} a function? Explain.
Is {(3, 1), (5, 2), (3, 4), (7, 6)} a function? Explain.
Table: x = {0, 1, 2, 3}, y = {4, 4, 4, 4}. Is this a function?
Table: x = {2, 4, 6, 2}, y = {1, 3, 5, 7}. Is this a function?
Write the domain and range of: {(1, 5), (2, 8), (3, 11), (4, 14)}.
Common Mistakes
Thinking a relation is a function just because it has ordered pairs — without checking for repeated x-values.
A relation is a function only if every x-value maps to exactly one y-value. Check all x-values.
Confusing the input (x) and output (y) — checking if y-values repeat instead of x-values.
For a function, x-values must be unique. Repeated y-values are fine.
Failing the vertical line test by drawing the line in the wrong direction.
The vertical line test uses vertical (up-down) lines. If any vertical line crosses the graph more than once, it's not a function.
Assuming every equation is a function — e.g., thinking x² + y² = 25 (a circle) is a function.
A circle fails the vertical line test. Not every equation defines y as a function of x.
Math Tips
One input → one output rule: Each x-value can only appear once in a function's table or mapping.
Repeated outputs are OK: Two different x-values can share the same y-value and still be a function.
Vertical Line Test: Draw (or imagine) vertical lines across the graph. If any line hits the graph twice, it is NOT a function.
Mapping diagrams: If any arrow from the input side splits into two outputs, it is NOT a function.
Sets of ordered pairs: Check if any x-value repeats with a different y-value. If so, not a function.