Domain & Range of Functions Cheat Sheet

Key Facts

• The domain of a function is the set of all possible input values, usually the allowed $x$-values.
• The range of a function is the set of all possible output values, usually the resulting $y$-values.
• A denominator cannot equal $0$, so for $f(x)=\frac{1}{x-a}$ the domain excludes $x=a$.
• For an even root such as $f(x)=\sqrt{x-a}$, the radicand must satisfy $x-a \ge 0$, so $x \ge a$.
• For a linear function $f(x)=mx+b$ with no stated restrictions, the domain is $(-\infty,\infty)$ and the range is $(-\infty,\infty)$ when $m \ne 0$.
• For a quadratic function $f(x)=a(x-h)^2+k$, the range is $y \ge k$ if $a>0$ and $y \le k$ if $a<0$.
• On a graph, read domain from the farthest left $x$-value to the farthest right $x$-value, and read range from the lowest $y$-value to the highest $y$-value.
• Use brackets $[\ ]$ when an endpoint is included and parentheses $(\ )$ when an endpoint is not included.

Vocabulary

Domain

The domain is the set of all input values that a function is allowed to use.

Range

The range is the set of all output values that a function can produce.

Interval Notation

Interval notation is a compact way to describe a set of numbers using endpoints, brackets, parentheses, and infinity symbols.

Restriction

A restriction is a value or condition that is not allowed because it would make the function undefined or impossible in context.

Undefined

An expression is undefined when it has no valid mathematical value, such as division by $0$.

Endpoint

An endpoint is a boundary value of an interval that may be included or excluded from the set.