A relation pairs inputs with outputs. The SAT may present the same relationship as ordered pairs, a table, a mapping, a formula, or a graph. The central question is whether each allowed input produces exactly one output.
Learning objectives
- Locate ordered pairs and quadrants on a coordinate plane.
- Identify domain and range from four common representations.
- Decide whether a relation is a function.
- Apply the vertical line test correctly.
- Evaluate function notation at numbers and algebraic expressions.
Coordinate plane and ordered pairs
The horizontal \(x\)-axis and vertical \(y\)-axis meet at the origin \((0,0)\). In \((x,y)\), move horizontally according to \(x\), then vertically according to \(y\). Quadrants proceed counterclockwise: I \((+,+)\), II \((-,+)\), III \((-,-)\), IV \((+,-)\). Points on an axis are not in a quadrant.
Relations, domain, and range
- Relation
- Any set of ordered pairs connecting inputs and outputs.
Domain versus range
Domain
All distinct input or \(x\)-values. Repeated inputs are listed once.
Range
All distinct output or \(y\)-values. Repeated outputs are listed once.
| Input x | Output y |
|---|---|
| \(-3\) | \(4\) |
| \(0\) | \(1\) |
| \(2\) | \(4\) |
| \(5\) | \(-2\) |
For the table, the domain is \(\{-3,0,2,5\}\) and the range is \(\{-2,1,4\}\). The repeated output \(4\) is written once in the range.
What makes a function
Test any representation
- Ordered pairs
Look for a repeated \(x\)-value with different \(y\)-values.
- Table
Scan the input column; any repeated input must keep the same output.
- Mapping
Every input node must have exactly one outgoing arrow.
- Graph
Apply the vertical line test.
Vertical line test
A graph represents \(y\) as a function of \(x\) if no vertical line intersects it more than once. The test counts outputs for one fixed input; horizontal lines do not answer that question.
Vertical line test examples
The dashed vertical line demonstrates how many outputs belong to one input.
Function notation and evaluation
The symbol \(f(x)\) names a function value; it does not mean \(f\times x\).
Substitute a grouped expression
Let \(f(x)=2x^2-3x+1\). Find \(f(a+1)\).
- Replace every x
Write \(2(a+1)^2-3(a+1)+1\).
- Expand the square
\((a+1)^2=a^2+2a+1\), so the first term is \(2a^2+4a+2\).
- Combine
\(2a^2+4a+2-3a-3+1=2a^2+a\).
Check your understanding
Which relation is not a function?
- \(\{(-1,2),(0,2),(3,5)\}\)
- \(\{(2,1),(2,4),(5,4)\}\)
- \(y=4\)
- \(y=|x|\)
Show answer and explanation
Answer: Choice B
Input \(2\) is paired with both \(1\) and \(4\). Shared outputs are allowed, but one input cannot have two outputs.
What to remember
- Domain contains inputs; range contains outputs.
- One input must have exactly one output for a function.
- Use repeated inputs, outgoing arrows, or vertical intersections according to representation.
- \(f(x)\) is an output label, not multiplication.
- Parenthesize a substituted expression everywhere \(x\) appears.
Put these notes into practice
Apply the ideas with SAT-style questions, then reinforce key details with flashcards.