AI Tutor
What happens when you enter 5÷05 \div 0 or −9\sqrt{-9} into a basic calculator? The screen flashes an error because those operations have no valid real-number answers.
The natural domain of a function is the complete set of real-number inputs (xx-values) that produce well-defined, real-number outputs (yy-values).
So which specific mathematical operations trigger these error traps?

The Two Red Flags

In real-variable algebra, domain restrictions arise almost entirely from two danger zones: dividing by zero and taking an even root of a negative number.
The quantity sitting underneath a radical symbol is called the radicand. For any even root (like x\sqrt{x} or x4\sqrt[4]{x}), that radicand must be non-negative: radicand≥0\text{radicand} \ge 0.
How do we apply the first rule when dealing with rational functions?

Rational Expressions: Excluding Zeros

A rational expression is a fraction where both numerator and denominator are polynomials. To find its domain, set the denominator equal to zero and exclude those roots.
For example, find the domain of f(x)=x+3x2−4x−5f(x) = \frac{x + 3}{x^2 - 4x - 5}. We set the denominator to zero: x2−4x−5=(x−5)(x+1)=0x^2 - 4x - 5 = (x - 5)(x + 1) = 0, giving x=5x = 5 and x=−1x = -1.