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.

$y = \sqrt3{x} - 4$

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

Explanation:

Step1: Calculate $\sqrt[3]{x}$ for $x=-8$

$\sqrt[3]{-8} = -2$

Step2: Calculate $y$ for $x=-8$

$y = -2 - 4 = -6$

Step3: Calculate $\sqrt[3]{x}$ for $x=-1$

$\sqrt[3]{-1} = -1$

Step4: Calculate $y$ for $x=-1$

$y = -1 - 4 = -5$

Step5: Calculate $\sqrt[3]{x}$ for $x=0$

$\sqrt[3]{0} = 0$

Step6: Calculate $y$ for $x=0$

$y = 0 - 4 = -4$

Step7: Calculate $\sqrt[3]{x}$ for $x=1$

$\sqrt[3]{1} = 1$

Step8: Calculate $y$ for $x=1$

$y = 1 - 4 = -3$

Step9: Calculate $\sqrt[3]{x}$ for $x=8$

$\sqrt[3]{8} = 2$

Step10: Calculate $y$ for $x=8$

$y = 2 - 4 = -2$

Step11: Identify domain of cube root function

Cube roots accept all real numbers, so domain is all real $x$.

Step12: Identify range of transformed function

Cube root outputs all real numbers; subtracting 4 shifts it, so range is all real $y$.

Completed Table:

$x$	$\sqrt[3]{x}$	$y = \sqrt[3]{x} - 4$
$-1$	$-1$	$-5$
$0$	$0$	$-4$
$1$	$1$	$-3$
$8$	$2$	$-2$

Domain and Range:

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

Answer:

Step1: Calculate $\sqrt[3]{x}$ for $x=-8$

$\sqrt[3]{-8} = -2$

Step2: Calculate $y$ for $x=-8$

$y = -2 - 4 = -6$

Step3: Calculate $\sqrt[3]{x}$ for $x=-1$

$\sqrt[3]{-1} = -1$

Step4: Calculate $y$ for $x=-1$

$y = -1 - 4 = -5$

Step5: Calculate $\sqrt[3]{x}$ for $x=0$

$\sqrt[3]{0} = 0$

Step6: Calculate $y$ for $x=0$

$y = 0 - 4 = -4$

Step7: Calculate $\sqrt[3]{x}$ for $x=1$

$\sqrt[3]{1} = 1$

Step8: Calculate $y$ for $x=1$

$y = 1 - 4 = -3$

Step9: Calculate $\sqrt[3]{x}$ for $x=8$

$\sqrt[3]{8} = 2$

Step10: Calculate $y$ for $x=8$

$y = 2 - 4 = -2$

Step11: Identify domain of cube root function

Cube roots accept all real numbers, so domain is all real $x$.

Step12: Identify range of transformed function

Cube root outputs all real numbers; subtracting 4 shifts it, so range is all real $y$.

Completed Table:

$x$	$\sqrt[3]{x}$	$y = \sqrt[3]{x} - 4$
$-1$	$-1$	$-5$
$0$	$0$	$-4$
$1$	$1$	$-3$
$8$	$2$	$-2$

Domain and Range:

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