Question

given the function below, fill in the table of values, use the table of values to graph the function, and then identify the function’s domain and range.

$y = \frac{1}{4}\sqrt3{x}$

$x$	$\sqrt3{x}$	$y = \frac{1}{4}\sqrt3{x}$
$-1$
$0$
$1$
$8$

Explanation:

Step1: Calculate cube roots for each $x$

For $x=-8$: $\sqrt[3]{-8} = -2$
For $x=-1$: $\sqrt[3]{-1} = -1$
For $x=0$: $\sqrt[3]{0} = 0$
For $x=1$: $\sqrt[3]{1} = 1$
For $x=8$: $\sqrt[3]{8} = 2$

Step2: Compute $y$ for each cube root

For $x=-8$: $y = \frac{1}{4} \times (-2) = -\frac{1}{2}$
For $x=-1$: $y = \frac{1}{4} \times (-1) = -\frac{1}{4}$
For $x=0$: $y = \frac{1}{4} \times 0 = 0$
For $x=1$: $y = \frac{1}{4} \times 1 = \frac{1}{4}$
For $x=8$: $y = \frac{1}{4} \times 2 = \frac{1}{2}$

Step3: Identify domain of the function

Cube roots are defined for all real numbers, so the domain is all real $x$.

Step4: Identify range of the function

Since cube roots produce all real numbers, scaling by $\frac{1}{4}$ also results in all real numbers, so the range is all real $y$.

Answer:

Completed Table:

$x$	$\sqrt[3]{x}$	$y = \frac{1}{4}\sqrt[3]{x}$
$-1$	$-1$	$-\frac{1}{4}$
$0$	$0$	$0$
$1$	$1$	$\frac{1}{4}$
$8$	$2$	$\frac{1}{2}$

Domain: All real numbers, or $(-\infty, \infty)$

Range: All real numbers, or $(-\infty, \infty)$