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.

$y = -\frac{1}{2}sqrt3{x - 4}$

$x$	$sqrt3{x - 4}$	$y = -\frac{1}{2}sqrt3{x - 4}$

answer
attempt 1 out of 2

Explanation:

Step1: Pick x-values for calculation

We choose convenient x-values: $x=-4, 4, 5, 0, 12$

Step2: Calculate $\sqrt[3]{x-4}$ for $x=-4$

Substitute $x=-4$:
$\sqrt[3]{-4 - 4} = \sqrt[3]{-8} = -2$

Step3: Calculate y for $x=-4$

Substitute $\sqrt[3]{x-4}=-2$:
$y = -\frac{1}{2} \times (-2) = 1$

Step4: Calculate $\sqrt[3]{x-4}$ for $x=4$

Substitute $x=4$:
$\sqrt[3]{4 - 4} = \sqrt[3]{0} = 0$

Step5: Calculate y for $x=4$

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

Step6: Calculate $\sqrt[3]{x-4}$ for $x=5$

Substitute $x=5$:
$\sqrt[3]{5 - 4} = \sqrt[3]{1} = 1$

Step7: Calculate y for $x=5$

Substitute $\sqrt[3]{x-4}=1$:
$y = -\frac{1}{2} \times 1 = -\frac{1}{2}$

Step8: Calculate $\sqrt[3]{x-4}$ for $x=0$

Substitute $x=0$:
$\sqrt[3]{0 - 4} = \sqrt[3]{-4} \approx -1.587$

Step9: Calculate y for $x=0$

Substitute $\sqrt[3]{x-4}\approx-1.587$:
$y = -\frac{1}{2} \times (-1.587) \approx 0.794$

Step10: Calculate $\sqrt[3]{x-4}$ for $x=12$

Substitute $x=12$:
$\sqrt[3]{12 - 4} = \sqrt[3]{8} = 2$

Step11: Calculate y for $x=12$

Substitute $\sqrt[3]{x-4}=2$:
$y = -\frac{1}{2} \times 2 = -1$

Step12: Identify domain

Cube roots accept all real numbers, so $x \in \mathbb{R}$

Step13: Identify range

Output of cube root is all real numbers, scaled by $-\frac{1}{2}$, so $y \in \mathbb{R}$

Answer:

Completed Table:

$x$	$\sqrt[3]{x-4}$	$y = -\frac{1}{2}\sqrt[3]{x-4}$
$4$	$0$	$0$
$5$	$1$	$-\frac{1}{2}$
$0$	$\sqrt[3]{-4} \approx -1.587$	$\approx 0.794$
$12$	$2$	$-1$

Domain and Range:

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