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
-------------------------------------------------------------------------
```
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
-------------------------------------------------------------------------
```
