The question how to find domain and range asks for two sets: the domain (all allowable inputs \(x\)) and the range (all attainable outputs \(y\)). In algebra, both are usually expressed in interval notation.
Domain Set of real \(x\)-values for which the function is defined.
Range Set of real \(y\)-values that the function can actually produce.
Common algebra rules for the domain
Most domain restrictions come from avoiding undefined operations or non-real values.
| Expression type | Restriction needed | Condition to enforce |
|---|---|---|
| Fraction \(\dfrac{p(x)}{q(x)}\) | Denominator cannot be zero | \(q(x)\ne 0\) |
| Even root \(\sqrt{g(x)}\), \(\sqrt[4]{g(x)}\), ... | Radicand must be nonnegative | \(g(x)\ge 0\) |
| Logarithm \(\log(g(x))\) | Argument must be positive | \(g(x)>0\) |
Worked example: find domain and range
Consider a representative function where both domain and range require reasoning:
\[ f(x)=\sqrt{x-2}. \]
Step A: Domain of \(f(x)=\sqrt{x-2}\)
The expression under the square root must be nonnegative:
\[ x-2\ge 0 \quad \Longrightarrow \quad x\ge 2. \]
Domain \[ [2,\infty). \]
Step B: Range of \(f(x)=\sqrt{x-2}\)
Let \(y=\sqrt{x-2}\). For real square roots, outputs are always nonnegative:
\[ y\ge 0. \]
To confirm that every \(y\ge 0\) is achievable, solve for \(x\) in terms of \(y\):
\[ y=\sqrt{x-2} \quad \Longrightarrow \quad y^2=x-2 \quad \Longrightarrow \quad x=y^2+2. \]
For any \(y\ge 0\), the value \(x=y^2+2\) satisfies \(x\ge 2\), so it lies in the domain. Therefore, every \(y\ge 0\) occurs.
Range \[ [0,\infty). \]
Visualization: domain and range as number lines
The first number line shows the domain \(x\ge 2\). The second number line shows the range \(y\ge 0\). Filled dots indicate included endpoints.
General checklist for how to find domain and range
The following procedure works for many algebra problems involving domain and range:
- Domain: list every restriction (no zero denominators, nonnegative even-root radicands, positive log arguments) and intersect them.
- Range: analyze outputs by solving \(y=f(x)\) for \(x\) when possible, using inequalities implied by squares and square roots, and checking which \(y\) values are attainable.
- Write both sets using interval notation (unions when there are excluded points or gaps).
Summary
Mastering how to find domain and range means (1) restricting inputs so every expression is defined over the real numbers, and (2) determining which outputs are produced by those allowed inputs, then expressing both sets clearly in interval notation.