Advanced Math / Nonlinear Functions

Radical and Rational Function Domains

Radical and rational functions often ask for domain restrictions, excluded inputs, and endpoint values. Square-root radicands must be nonnegative, and denominators cannot equal zero.

Least you need to know

For $\sqrt{g(x)}$, require $g(x)\ge0$.
For $\frac{p(x)}{q(x)}$, require $q(x)\ne0$.
The numerator of a rational function may equal zero.
When asked for least or greatest integer values in a domain, solve the domain condition first.

Key notation

$f(x)=\sqrt{2x-6}$ — A radical function with domain $x\ge3$
$r(x)=\frac{x+4}{x^2-9}$ — A rational function excluding $x=3$ and $x=-3$

Worked example

For $f(x)=\sqrt{2x-6}$, require $2x-6\ge0$.
Then $2x\ge6$, so $x\ge3$.
The least integer in the domain is $3$.

Common mistakes

A radicand may equal zero.
A denominator may not equal zero.
If a denominator factors, every zero of every denominator factor is excluded.

How to recognize it

The problem asks for the domain or an excluded input.
The function has a square root or denominator involving $x$.
The problem asks for a least or greatest integer in the domain.

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