# Finding the Domain and Range of a Function

Read Domain and Range of a Function from the Algebra I curriculum for a basic introduction to domain and range.

## Finding the Domain of a Function from a Formula

The following situations are not allowed, so you must exclude value(s) that cause:

- Division by zero
- Even roots of negative numbers
- Zero raised to the zero power

## Example: Find the Domain of $\,g(x) = \frac{\sqrt{x-3}}{x-5}$

SOLUTION:

$$ \begin{gather} \cssId{s11}{x-3\not\lt 0\ \text{ and }\ x-5\ne 0}\cr\cr \cssId{s12}{x-3\ge 0 \ \text{ and }\ x-5\ne 0}\cr\cr \cssId{s13}{x\ge 3\ \text{ and }\ x\ne 5}\cr\cr \cssId{s14}{\text{dom}(g) = [3,5) \cup (5,\infty)} \end{gather} $$## Finding the Domain of a Function from a Graph

‘Collapse’ each point into its $x$-value.

## Finding the Range of a Function

The range of a function is its output set. It's easiest to find the range when you have the graph of the function. ‘Collapse’ each point into its $y$-value.

## Example: Find the Range of $\,f(x) = |x-5| + 3$

SOLUTION: Graph $\,f\,$ by starting with $\,y = |x|\,,$ shifting it right $\,5\,$ and up $\,3\,.$

$$ \cssId{s27}{\text{ran}(f) = [3,\infty)} $$