Question 3

Consider continuous functions [tex]f[/tex] and [tex]g[/tex]. Then complete the statement.

\begin{tabular}{|c|c|}
\hline
[tex]$x$[/tex] & [tex]$f(x)$[/tex] \\
\hline
0 & \\
\hline
-1 & -4 \\
\hline
0 & -2 \\
\hline
3 & 0 \\
\hline
8 & 2 \\
\hline
15 & 4 \\
\hline
24 & 6 \\
\hline
\end{tabular}

Function [tex]g[/tex] is the sum of 2 and the cube root of the sum of three times [tex]x[/tex] and 1.

Select the correct answer from each drop-down.

The [tex]$x$[/tex]-intercept of function [tex]f[/tex] is [tex]$\square$[/tex] the [tex]$x$[/tex]-intercept of function [tex]g[/tex].



Answer :

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].