Answer :
From the given table of values for the function, we can analyze how the function behaves:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -4 & -0.906 \\ \hline -2 & -0.625 \\ \hline 0 & 0.5 \\ \hline 2 & 5 \\ \hline 4 & 23 \\ \hline \end{array} \][/tex]
### Step-by-Step Analysis:
1. Examining the y-values for changes in sign:
- When [tex]\( x = -4 \)[/tex], [tex]\( y = -0.906 \)[/tex] (y is negative)
- When [tex]\( x = -2 \)[/tex], [tex]\( y = -0.625 \)[/tex] (y is negative)
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 0.5 \)[/tex] (y is positive)
2. Trend of y-values:
- As [tex]\( x \)[/tex] increases from [tex]\(-4\)[/tex] to [tex]\(-2\)[/tex] to [tex]\(0\)[/tex] to [tex]\(2\)[/tex] to [tex]\(4\)[/tex], the y-values are increasing: [tex]\(-0.906, -0.625, 0.5, 5, 23\)[/tex].
3. X-intercept analysis:
- The sign of the y-value changes from negative to positive between [tex]\( x = -2 \)[/tex] and [tex]\( x = 0 \)[/tex]. This indicates that the function crosses the x-axis between these points.
- Given that the sign change happens once and there are no other sign changes observed in the provided data, it suggests the function has exactly one x-intercept between [tex]\( x = -2 \)[/tex] and [tex]\( x = 0 \)[/tex].
Based on these observations, the correct conclusion is:
A. the function has exactly one [tex]\( x \)[/tex]-intercept
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -4 & -0.906 \\ \hline -2 & -0.625 \\ \hline 0 & 0.5 \\ \hline 2 & 5 \\ \hline 4 & 23 \\ \hline \end{array} \][/tex]
### Step-by-Step Analysis:
1. Examining the y-values for changes in sign:
- When [tex]\( x = -4 \)[/tex], [tex]\( y = -0.906 \)[/tex] (y is negative)
- When [tex]\( x = -2 \)[/tex], [tex]\( y = -0.625 \)[/tex] (y is negative)
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 0.5 \)[/tex] (y is positive)
2. Trend of y-values:
- As [tex]\( x \)[/tex] increases from [tex]\(-4\)[/tex] to [tex]\(-2\)[/tex] to [tex]\(0\)[/tex] to [tex]\(2\)[/tex] to [tex]\(4\)[/tex], the y-values are increasing: [tex]\(-0.906, -0.625, 0.5, 5, 23\)[/tex].
3. X-intercept analysis:
- The sign of the y-value changes from negative to positive between [tex]\( x = -2 \)[/tex] and [tex]\( x = 0 \)[/tex]. This indicates that the function crosses the x-axis between these points.
- Given that the sign change happens once and there are no other sign changes observed in the provided data, it suggests the function has exactly one x-intercept between [tex]\( x = -2 \)[/tex] and [tex]\( x = 0 \)[/tex].
Based on these observations, the correct conclusion is:
A. the function has exactly one [tex]\( x \)[/tex]-intercept
