Domain of a root of a fraction

Problem

Determine the natural domain of $f(x) = \sqrt{\dfrac{x+2}{x-5}}$.

Solution

The even-index root requires the radicand to be $\ge 0$ and the denominator non-zero: $\frac{x+2}{x-5}\ge 0, \qquad x\ne 5.$ Sign study of the fraction: numerator $x+2\ge 0 \iff x\ge -2$; denominator $x-5>0 \iff x>5$. The fraction is $\ge 0$ when numerator and denominator have the same sign (or the numerator is zero):

• $x\le -2$: numerator $\le 0$, denominator $<0$ → quotient $\ge 0$ ✓
• $-2<x<5$: numerator $>0$, denominator $<0$ → quotient $<0$ ✗
• $x>5$: both positive → quotient $>0$ ✓

Domain: $\boxed{(-\infty;-2]\cup(5;+\infty)}$.

Links

Topics: Functions and properties
Concepts: Natural domain
Skills: Solving inequalities · Studying a function
Exercise types: Study of a function