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.
درس پیشنهادی بعدی
این موضوع را با Extraneous solutions and domain checks ادامه دهید.
Extraneous solutions and domain checksدرسهای مرتبط
با درسهای نزدیک در همان موضوع ادامه دهید.