Functions: Range

The range of a function is the set of all possible output values (usually $y$-values) obtained by evaluating the function over its entire domain.

For the principal square root function $f(x) = \sqrt{x}$, the output is defined as the non-negative square root. Since we cannot obtain a negative number from a real square root, the range is restricted to non-negative numbers: $y \ge 0$ (or $[0, \infty)$ in interval notation).

A quadratic function like $f(x) = x^2$ takes any input and multiplies it by itself. Because the product of any real number with itself is never negative (e.g., $(-3)^2 = 9$ and $0^2 = 0$), the domain is all real numbers ($\mathbb{R}$), but the outputs are always non-negative. Therefore, its range is $y \ge 0$ (or $[0, \infty)$).

Periodic functions repeat their values in cycles. For these functions, the output values cycle between a specific minimum and maximum value, meaning their range is bounded and forms a closed interval: $[y_{\text{min}}, y_{\text{max}}]$ (or $y_{\text{min}} \le y \le y_{\text{max}}$). For example, the sine function, $f(x) = \sin(x)$, cycles continuously, restricting its range to $[-1, 1]$ (don't worry if you aren't familiar with $\sin(x)$ yet—it is only used here as an example).

When answering range questions, you can enter inequalities directly using keyboard shortcuts. The editor automatically converts them to math symbols: