Question

Find the domain and range of the function without graphing. Explain how you found your answers. $y=\frac{1}{3}\sqrt{x }-4$

Answer 1

$y=\frac{2}{3}\sqrt{x}-4$ is an increasing function of x. As x increases the function increases.

Domain is a set of x values where the function is real. The function is real if and only if $x\ge0$ since the square root of a negative number is a complex number. Thus, the dormain is $[0,\infin)$

Range is a set of values corresponding to the dormain. Since the function is an increasing function of x, plugging in the Lower and upper bounds of the dormain results in the Lower and upper bounds of the range respectively.

$f(0)=\frac{2}{3}\sqrt{0}-4 =-4$ and

$f(\infin)=\infin$

Thus, the range is $[-4,\infin)$