The Square-Root Function

Definition For a real number $a \ge 0$, the principal square root $\sqrt{a}$ is the unique real number $b$ with

The principal square-root function is $f(x) = \sqrt{x}$ for $x \ge 0$. A positive number $a$ has two real square roots, $\sqrt{a}$ and $-\sqrt{a}$, and the symbol always means the nonnegative one. As a consequence, $\sqrt{x^2} = \lvert x \rvert$ for every real $x$. The graph of $y = \sqrt{x}$ is only the upper half of the sideways parabola $x = y^2$, the half with $y \ge 0$.

Definition The radicand is the expression under the radical sign. In $\sqrt{6 - x}$ the radicand is $6 - x$. In the real numbers a square root is defined exactly when its radicand is greater than or equal to $0$.

Definition The domain of a function given by a formula, when no domain is stated, is the set of all real inputs for which the formula gives a real number. For a square root formula $\sqrt{u(x)}$ that means the solution set of $u(x) \ge 0$, together with any conditions $u$ itself needs.

Definition The range of a function is the set of all its outputs. The range of $f(x) = \sqrt{x}$ is $[0, \infty)$. Every output is nonnegative by definition, and every $y \ge 0$ is actually reached, at the input $x = y^2$, since $\sqrt{y^2} = y$ when $y \ge 0$.

Definition An endpoint of a graph is a point $(c, f(c))$ where $c$ is in the domain and is an end of an interval that makes up the domain, so near $x = c$ the graph has points on one side and none on the other. For $f(x) = \sqrt{u(x)} + k$ the domain is where $u(x) \ge 0$, and its ends are found among the inputs where the radicand $u(x)$ is $0$. In the common case of a linear radicand $u(x) = mx + b$ with $m \ne 0$ there is exactly one endpoint, at the input $c = -\frac{b}{m}$, with output $\sqrt{0} + k = k$. The domain is the closed half-line $[c, \infty)$ when $m > 0$ and $(-\infty, c]$ when $m < 0$. A zero of the radicand is not always an endpoint. The radicand of $\sqrt{x^2}$ is $0$ at $x = 0$, yet the domain is all of $\mathbb{R}$ and the graph continues on both sides.

Definition A coordinate pair $(a, b)$ lies on the graph of a function $f$ exactly when $a$ is in the domain of $f$ and $f(a) = b$. For a square root formula, exact pairs are easiest to find by choosing the radicand first, using perfect squares $0$, $1$, $4$, $9$, $16$, and then solving for the input that gives that radicand.

Definition A function $f$ is increasing on an interval $I$ if $f(a) < f(b)$ whenever $a$ and $b$ are in $I$ with $a < b$. The square-root function is increasing on $[0, \infty)$. Suppose $0 \le a < b$. If $\sqrt{a} \ge \sqrt{b}$ held, then squaring, which preserves order among nonnegative numbers, would give $a \ge b$, a contradiction. So

Definition A function $f$ is decreasing on an interval $I$ if $f(a) > f(b)$ whenever $a$ and $b$ are in $I$ with $a < b$. The function $g(x) = \sqrt{6 - x}$ is decreasing on $(-\infty, 6]$. If $a < b \le 6$, then $6 - a > 6 - b \ge 0$, and since the square root is increasing, $\sqrt{6 - a} > \sqrt{6 - b}$.