Sketch the graph of $f(x) = \sqrt{x}$. Find the domain and range.

The square root function $f(x) = \sqrt{x}$ is defined as the non-negative square root of $x$, and it exists only for non-negative values of $x$.

Domain

Since we cannot take the square root of a negative number (in real numbers):

$$\text{Domain} = [0, \infty) = \{x \in \mathbb{R} : x \geq 0\}$$

Range

Since $\sqrt{x}$ always gives a non-negative output:

$$\text{Range} = [0, \infty) = \{y \in \mathbb{R} : y \geq 0\}$$

Sketch

The graph is a smooth curve starting from the origin, bending gently to the right.

Conclusion: The square root function is a one-to-one, continuous, and increasing function defined for all $x \geq 0$, with output values also $\geq 0$.