Functions & Relations Cheat Sheet

A printable reference covering relations, functions, domain, range, function notation, graphs, mappings, and the vertical line test for grades 8-10.

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Functions and relations describe how inputs are connected to outputs using tables, graphs, equations, ordered pairs, and mapping diagrams. This cheat sheet helps students decide whether a relation is a function and how to identify domain and range. It is useful for algebra, coordinate graphing, and interpreting real-world patterns. Students need these ideas before working with linear, quadratic, exponential, and other function families. The most important rule is that each input can have exactly one output in a function. Function notation such as $f(x)$ names the output produced by the input $x$ . Domain means all allowed input values, and range means all possible output values. Graphs, tables, and equations all show the same relationship in different forms.

Key Facts

A relation is any set of ordered pairs $(x,y)$ that connects inputs $x$ to outputs $y$ .
A function is a relation where every input $x$ has exactly one output $y$ .
Function notation $f(x)$ means the output of the function $f$ when the input is $x$ .
The domain is the set of all possible input values $x$ , and the range is the set of all possible output values $y$ .
A graph represents a function if every vertical line $x=a$ crosses the graph at no more than one point.
In an equation such as $y=2x+3$ , the input is usually $x$ and the output is usually $y$ .
To evaluate a function, substitute the input value into the rule, such as $f(4)=2(4)+3=11$ .
A linear function has the form $f(x)=mx+b$ , where $m$ is the slope and $b$ is the $y$ -intercept.

Vocabulary

Relation: A relation is a pairing of input values with output values, often written as ordered pairs $(x,y)$ .
Function: A function is a relation in which each input value has exactly one output value.
Domain: The domain is the set of all input values that can be used in a relation or function.
Range: The range is the set of all output values produced by a relation or function.
Function Notation: Function notation, such as $f(x)$ , names the output of a function for the input $x$ .
Vertical Line Test: The vertical line test says a graph is a function only if every vertical line intersects it at most once.

Common Mistakes to Avoid

Repeating an input with different outputs, such as $(2,3)$ and $(2,5)$ , is wrong for a function because the input $2$ has two outputs.
Confusing domain and range is wrong because domain lists input values $x$ , while range lists output values $y$ .
Treating $f(x)$ as multiplication is wrong because $f(x)$ means the value of the function at input $x$ , not $f \cdot x$ .
Using the vertical line test sideways is wrong because vertical lines test whether one $x$ -value gives more than one $y$ -value.
Forgetting to substitute the input everywhere is wrong because in $f(x)=x^2+2x$ , evaluating $f(3)$ requires $3^2+2(3)$ .

Practice Questions

1 For the relation $\{(1,4),(2,5),(3,6),(2,7)\}$ , determine whether it is a function and explain why.
2 Given $f(x)=3x-2$ , find $f(5)$ and $f(-1)$ .
3 For the ordered pairs $\{(-2,1),(0,1),(4,3),(6,3)\}$ , list the domain and the range.
4 Explain why a circle does not pass the vertical line test, even though it is a graph made of ordered pairs.