Question

which statement best describes $g(x) = \sqrt3{x + 6} - 8$ and the parent function $f(x) = \sqrt3{x}$?

• the domains of $g(x)$ and $f(x)$ are the same, but their ranges are not the same.
• the ranges of $g(x)$ and $f(x)$ are the same, but their domains are not the same.
• the ranges of $g(x)$ and $f(x)$ are the same, and their domains are also the same.
• 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 Cube Root Functions

For a cube root function \( y = \sqrt[3]{u} \), the expression inside the cube root (\( u \)) can be any real number because the cube of a negative number is negative, the cube of zero is zero, and the cube of a positive number is positive. So for \( f(x)=\sqrt[3]{x} \), the domain is all real numbers (\( (-\infty, \infty) \)) since \( x \) can be any real number. For \( g(x)=\sqrt[3]{x + 6}-8 \), the expression inside the cube root is \( x + 6 \). Since \( x+6 \) is a linear expression and \( x \) can be any real number (making \( x + 6 \) any real number), the domain of \( g(x) \) is also all real numbers (\( (-\infty, \infty) \)).

Step2: Analyze Range of Cube Root Functions

The range of a cube root function \( y=\sqrt[3]{u} \) is all real numbers because for any real number \( y \), we can find a real number \( u \) such that \( y = \sqrt[3]{u} \) (specifically, \( u = y^3 \)). For \( f(x)=\sqrt[3]{x} \), as \( x \) takes all real values, \( f(x) \) takes all real values (range \( (-\infty, \infty) \)). For \( g(x)=\sqrt[3]{x + 6}-8 \), let \( u=x + 6 \). The function \( \sqrt[3]{u} \) has a range of all real numbers, and subtracting 8 from a function that can take any real value just shifts the graph vertically, but the set of all possible output values (range) still remains all real numbers. So the range of \( g(x) \) is also \( (-\infty, \infty) \).

Answer:

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