To answer how to find the domain of a function, interpret “domain” as the set of all real input values \(x\) for which the function produces a real output. In algebra, domain restrictions usually come from operations that can become undefined or non-real.
Definition The domain of a function is the set of all real numbers \(x\) such that every expression appearing in the function is defined (over the real numbers).
Core rules for domain restrictions
Many functions are defined for all real numbers, but certain algebraic forms impose restrictions. The most common cases are listed below.
| Function feature | What can go wrong | Required condition for the domain |
|---|---|---|
| Fraction \(\dfrac{p(x)}{q(x)}\) | Division by zero | \(q(x)\ne 0\) |
| Even root \(\sqrt[n]{g(x)}\) with even \(n\) | Non-real output | \(g(x)\ge 0\) |
| Logarithm \(\log(g(x))\) | Log of non-positive number | \(g(x)>0\) |
| Odd root \(\sqrt[3]{g(x)}\), \(\sqrt[5]{g(x)}\), ... | No real-number issue | All real \(x\) (unless other restrictions appear) |
Worked example combining common restrictions
Since the keyword is general, consider a representative algebraic function that contains a fraction, a square root, and a logarithm:
\[ f(x)=\frac{\sqrt{x-2}}{x-5}+\ln(x+1). \]
Finding the domain means enforcing every restriction simultaneously.
Step 1: Square root restriction
The radicand must be nonnegative:
\[ x-2\ge 0 \quad \Longrightarrow \quad x\ge 2. \]
Step 2: Denominator restriction
The denominator must not be zero:
\[ x-5\ne 0 \quad \Longrightarrow \quad x\ne 5. \]
Step 3: Logarithm restriction
The log argument must be positive:
\[ x+1>0 \quad \Longrightarrow \quad x>-1. \]
Step 4: Combine all conditions
All restrictions must hold at the same time. The strongest lower bound is \(x\ge 2\) (which automatically satisfies \(x>-1\)), and the value \(x=5\) must be excluded.
Domain \[ \text{Domain}=[2,5)\cup(5,\infty). \]
The endpoint \(2\) is included because \(\sqrt{x-2}\) is defined at \(x=2\). The point \(x=5\) is excluded because it makes the denominator zero.
Visualization: domain on a number line
The number line below represents the domain for the example: all real numbers from \(2\) onward, except \(x=5\). A filled dot means included; an open circle means excluded.
General procedure for how to find the domain of a function
The same logic applies to most algebraic functions:
- Identify any denominators and require them to be nonzero.
- Identify any even roots and require their radicands to be nonnegative.
- Identify any logarithms and require their arguments to be positive.
- Combine all restrictions by intersection, then write the domain in interval notation.
Summary
The key to how to find the domain of a function is enforcing real-number validity: exclude zeros of denominators, require nonnegative radicands for even roots, and require positive arguments for logarithms, then express the remaining \(x\)-values in interval notation.