Relations vs. Functions
Every function is a relation, but not every relation is a function. The distinction sets the foundation for all of Pre-Calculus.
Introduction
Prerequisite Connection: In Algebra, you learned to solve equations like by substituting values—this lesson formalizes that intuition by defining when such input-output relationships qualify as functions.
Today's Increment: We define a function as a special type of relation where each input has exactly one output. You'll master the Vertical Line Test and learn to read mapping diagrams to distinguish functions from non-functions.
Why This Matters for Calculus: Calculus is built entirely on functions—derivatives and integrals only make sense when each input produces a unique output. Without this foundational concept, the limit would be ambiguous.
Explanation of Key Concepts
Relation
A relation is any set of ordered pairs . There are no restrictions on how many outputs each input can have.
Function
A function is a relation where each input (-value) maps to exactly one output (-value).
The Vertical Line Test
Each vertical line represents a single -value.
Count how many times the line crosses the graph.
Every vertical line crosses at most once
Any vertical line crosses more than once
Reading Mapping Diagrams
Each input has exactly one arrow leaving it
Input 1 has TWO arrows → violates function rule
Worked Examples
Example 1: Ordered Pairs
Determine if the relation is a function.
List the -values (first coordinates):
Does any -value appear more than once? No.
Each input (1, 2, 3, 4) appears exactly once, with unique outputs (3, 5, 7, 9).
Yes, this is a function. Each input has exactly one output.
Example 2: Vertical Line Test on a Circle
Is the equation a function of ?
This is the equation of a circle centered at the origin with radius .
Consider the vertical line . Substitute into the equation:
The vertical line at intersects the circle at two points: and .
No, this is NOT a function. The input produces two outputs.
Example 3: Implicit Relations with Parameters
For what values of the constant does the relation define as a function of ?
This notation means two branches: and .
For any (when ), both and exist, giving two -values for one -value.
The full relation is never a function for any non-zero . However, we can restrict to one branch: (upper) or (lower).
Calculus Connection: This "branch selection" is exactly what we do with inverse functions (Lesson 14) and is essential when computing via implicit differentiation.
No value of makes the full relation a function. You must restrict to a single branch.
Common Pitfalls
Watch Out For These Mistakes
- Confusing "same output" with "same input": Two different inputs can share the same output (e.g., and ). That's allowed! The rule only bans one input from having multiple outputs.
- Using the Horizontal Line Test too early: The Horizontal Line Test checks if a function is one-to-one (injective). It does NOT determine if a relation is a function—that's the Vertical Line Test.
- Ignoring implicit equations: Equations like don't explicitly solve for . Always solve for or apply the Vertical Line Test graphically.
Real-World Application
ATM Transactions as Functions
When you swipe your debit card (input: card number), the ATM returns your account balance (output: dollar amount).
This is a function: One card number always maps to exactly one balance. If the same card returned multiple different balances simultaneously, the banking system would be broken!
What's NOT a function: Asking "Which card numbers have a balance of $500?" Multiple people could have $500—so the balance doesn't uniquely determine the card. This reverse mapping is a relation, not a function.
Practice Quiz
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