Given the values and the definition of function [tex]\( g \)[/tex], we need to find the [tex]\( x \)[/tex]-intercept of [tex]\( g \)[/tex].
The function [tex]\( g(x) \)[/tex] is defined as the sum of 2 and the cube root of the sum of three times [tex]\( x \)[/tex] and 1. That is:
[tex]\[ g(x) = 2 + \sqrt[3]{3x + 1} \][/tex]
To find the [tex]\( x \)[/tex]-intercept of [tex]\( g \)[/tex], we need to solve for [tex]\( x \)[/tex] when [tex]\( g(x) = 0 \)[/tex]:
[tex]\[ 0 = 2 + \sqrt[3]{3x + 1} \][/tex]
First, isolate the cube root term by subtracting 2 from both sides:
[tex]\[ -2 = \sqrt[3]{3x + 1} \][/tex]
Next, cube both sides to remove the cube root:
[tex]\[ (-2)^3 = 3x + 1 \][/tex]
This simplifies to:
[tex]\[ -8 = 3x + 1 \][/tex]
Subtract 1 from both sides to isolate terms with [tex]\( x \)[/tex]:
[tex]\[ -9 = 3x \][/tex]
Divide both sides by 3 to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{-9}{3} \][/tex]
[tex]\[ x = -3 \][/tex]
Thus, the [tex]\( x \)[/tex]-intercept of the function [tex]\( g \)[/tex] is [tex]\( x = -3 \)[/tex].
The problem does not provide specific information about the [tex]\( x \)[/tex]-intercept of function [tex]\( f \)[/tex], so we cannot make a direct comparison. However, we do know the [tex]\( x \)[/tex]-intercept for [tex]\( g \)[/tex].
Based on the obtained solution, the correct answer to the statement is:
The [tex]\( x \)[/tex]-intercept of function [tex]\( f \)[/tex] is unknown, and the [tex]\( x \)[/tex]-intercept of function [tex]\( g \)[/tex] is [tex]\(-3\)[/tex].
