Compare continuous functions [tex]\( f, g \)[/tex], and [tex]\( h \)[/tex], and match the statements with the function they best describe. Drag each description to the correct location on the table.

\begin{tabular}{|c|c|c|c|}
\hline
& [tex]\( x \)[/tex] & [tex]\( |g(x)| \)[/tex] & \\
\hline
& -2 & 8 & \multirow{6}{*}{\begin{tabular}{l}
Function [tex]\( h \)[/tex] is the \\
sum of 3 and \\
four times the \\
cube of [tex]\( x \)[/tex].
\end{tabular}} \\
\hline
& -1 & 0.5 & \\
\hline
& 0 & -1 & \\
\hline
& 1 & -2.5 & \\
\hline
\end{tabular}

- This function is decreasing over the longest interval.
- This function has the lowest [tex]\( y \)[/tex]-intercept.
- This function has the highest [tex]\( y \)[/tex]-intercept.
- This function is increasing over the longest interval.



Answer :

To solve this problem, let's carefully analyze the descriptions of the functions and the given values.

First, we need to identify each function based on given characteristics and values.

### Functions Analyzed

1. Function [tex]\( g(x) \)[/tex]
- Given values of [tex]\( |g(x)| \)[/tex] at different points:
- [tex]\( x = -2 \rightarrow |g(x)| = 8 \)[/tex]
- [tex]\( x = -1 \rightarrow |g(x)| = 0.5 \)[/tex]
- [tex]\( x = 0 \rightarrow |g(x)| = -1 \)[/tex]
- [tex]\( x = 1 \rightarrow |g(x)| = -2.5 \)[/tex]
- Note: Since the absolute value function, there could be some clarifying points needed here.

2. Function [tex]\( h(x) \)[/tex]
- The description is "Function [tex]\( h \)[/tex] is the sum of 3 and four times the cube of [tex]\( x \)[/tex]."
- This means: [tex]\( h(x) = 3 + 4x^3 \)[/tex]

### Match Descriptions:

1. This function is decreasing over the longest interval:
- Analysis: A function that decreases over its entire domain or for a large part of it would fit this description.
- Since [tex]\( g(x) \)[/tex] has negative values and squares, more contextual detailing should help. But we know cubic functions tend to have local minima and maxima.

2. This function has the lowest [tex]\( y \)[/tex]-intercept:
- Analysis: Identify the y-intercepts of the functions. :
- The y-intercept of [tex]\( h(x) = 3 + 4x^3 \)[/tex] is [tex]\( h(0) = 3 \)[/tex].

3. This function has the highest [tex]\( y \)[/tex]-intercept:
- Compare the y-intercepts seen briefly, as previous variable-based lookups show.

4. This function is increasing over the longest interval:
- Analysis: The cubic function [tex]\( h(x) = 3 + 4x^3 \)[/tex] increases indefinitely for positive and negative values of [tex]\( x \)[/tex].

### Matching Statements:

- This function is decreasing over the longest interval:
- For further analyzing, since h(x) has been shown generic increasing slopes left, the component g(x) indicates potential decreases.

- This function has the lowest [tex]\( y \)[/tex]-intercept:
- Cross-check other intercepts w.r.t lesser values, verifies aligning calculations.

- This function has the highest [tex]\( y \)[/tex]-intercept:
- Seen intercept/h(x) "highest y-intercept" is three.

- This function is increasing over the longest interval:
- Matches function centric data analysis.

Finally, the matches are detailed for clear structural understanding.

```plaintext
-------------------------------------------------------------------------
Description | Matched Function
-------------------------------------------------------------------------
This function is decreasing over the largest interval | g(x) check-in necessary.
-------------------------------------------------------------------------
This function has the lowest y-intercept | lowest-to-value h(x)
-------------------------------------------------------------------------
This function has the highest y-intercept | Highest clarity re h(x)
-------------------------------------------------------------------------
This function is increasing over the longest interval | Valid, generic alignment to potential maximum domains
-------------------------------------------------------------------------
```