Question

which statement best describes $g(x) = \sqrt3{x + 6} - 8$ and the parent function $f(x) = \sqrt3{x}$?
\bigcirc the domains of $g(x)$ and $f(x)$ are the same, but their ranges are not the same.
\bigcirc the ranges of $g(x)$ and $f(x)$ are the same, but their domains are not the same.
\bigcirc the ranges of $g(x)$ and $f(x)$ are the same, and their domains are also the same.
\bigcirc the domains of $g(x)$ and $f(x)$ are the not the same, and their ranges are also not the same.

Explanation:

Step1: Analyze Domain of \( f(x) \)

The parent function is \( f(x)=\sqrt[3]{x} \). The cube root function is defined for all real numbers (since we can take the cube root of any real number, positive, negative, or zero). So, the domain of \( f(x) \) is \( (-\infty, \infty) \).

Step2: Analyze Domain of \( g(x) \)

For \( g(x)=\sqrt[3]{x + 6}-8 \), we look at the expression inside the cube root, which is \( x + 6 \). Since we can add 6 to any real number \( x \), and then take the cube root, the domain of \( g(x) \) is also all real numbers. So, domain of \( g(x) \) is \( (-\infty, \infty) \).

Step3: Analyze Range of \( f(x) \)

The cube root function \( f(x)=\sqrt[3]{x} \) can output any real number. If \( x \) is positive, \( \sqrt[3]{x} \) is positive; if \( x \) is negative, \( \sqrt[3]{x} \) is negative; and if \( x = 0 \), \( \sqrt[3]{x}=0 \). So, the range of \( f(x) \) is \( (-\infty, \infty) \).

Step4: Analyze Range of \( g(x) \)

For \( g(x)=\sqrt[3]{x + 6}-8 \), let \( y=\sqrt[3]{x + 6}-8 \). We can solve for \( x \) in terms of \( y \): \( y + 8=\sqrt[3]{x + 6} \), then \( (y + 8)^3=x + 6 \), and \( x=(y + 8)^3-6 \). For any real number \( y \), we can find a corresponding real number \( x \). So, the range of \( g(x) \) is also all real numbers, \( (-\infty, \infty) \).

Step5: Compare Domains and Ranges

Since both the domain of \( f(x) \) and \( g(x) \) is \( (-\infty, \infty) \), and both the range of \( f(x) \) and \( g(x) \) is \( (-\infty, \infty) \), the ranges and domains are the same.

Answer:

The ranges of \( g(x) \) and \( f(x) \) are the same, and their domains are also the same.