Answered

\begin{tabular}{|c|c|}
\hline
[tex]$x$[/tex] & [tex]$I(x)$[/tex] \\
\hline
-4 & -221 \\
\hline
-3 & 140 \\
\hline
-2 & 231 \\
\hline
-1 & 160 \\
\hline
0 & 35 \\
\hline
1 & -36 \\
\hline
2 & 55 \\
\hline
3 & 416 \\
\hline
\end{tabular}

Which describes the location of the zeroes of the function?

A. Between [tex]$x=-4$[/tex] and [tex]$x=-3$[/tex], between [tex]$x=0$[/tex] and [tex]$x=1$[/tex], and between [tex]$x=1$[/tex] and [tex]$x=2$[/tex].
B. Between [tex]$x=-1$[/tex] and [tex]$x=0$[/tex] and between [tex]$x=0$[/tex] and [tex]$x=1$[/tex].
C. Between [tex]$x=35$[/tex] and [tex]$x=160$[/tex] and between [tex]$x=-36$[/tex] and [tex]$x=35$[/tex].
D. Between [tex]$x=-221$[/tex] and [tex]$x=140$[/tex], between [tex]$x=-36$[/tex] and [tex]$x=35$[/tex], and between [tex]$x=55$[/tex] and [tex]$x=416$[/tex].



Answer :

GinnyAnswer
To determine the locations of the zeroes of the function [tex]\( I(x) \)[/tex], we need to look at the given data points and observe where the function value changes sign. The changes in sign indicate that the function crosses the x-axis between those points.

Let's analyze the data points step-by-step:

1. For [tex]\( x = -4 \)[/tex] and [tex]\( x = -3 \)[/tex]:
- [tex]\( I(-4) = -221 \)[/tex] and [tex]\( I(-3) = 140 \)[/tex].
- The function changes from negative to positive, indicating a zero between [tex]\( x = -4 \)[/tex] and [tex]\( x = -3 \)[/tex].

2. For [tex]\( x = -3 \)[/tex] and [tex]\( x = -2 \)[/tex]:
- [tex]\( I(-3) = 140 \)[/tex] and [tex]\( I(-2) = 231 \)[/tex].
- Both values are positive, no zero crossing here.

3. For [tex]\( x = -2 \)[/tex] and [tex]\( x = -1 \)[/tex]:
- [tex]\( I(-2) = 231 \)[/tex] and [tex]\( I(-1) = 160 \)[/tex].
- Both values are positive, no zero crossing here.

4. For [tex]\( x = -1 \)[/tex] and [tex]\( x = 0 \)[/tex]:
- [tex]\( I(-1) = 160 \)[/tex] and [tex]\( I(0) = 35 \)[/tex].
- Both values are positive, no zero crossing here.

5. For [tex]\( x = 0 \)[/tex] and [tex]\( x = 1 \)[/tex]:
- [tex]\( I(0) = 35 \)[/tex] and [tex]\( I(1) = -36 \)[/tex].
- The function changes from positive to negative, indicating a zero between [tex]\( x = 0 \)[/tex] and [tex]\( x = 1 \)[/tex].

6. For [tex]\( x = 1 \)[/tex] and [tex]\( x = 2 \)[/tex]:
- [tex]\( I(1) = -36 \)[/tex] and [tex]\( I(2) = 55 \)[/tex].
- The function changes from negative to positive, indicating a zero between [tex]\( x = 1 \)[/tex] and [tex]\( x = 2 \)[/tex].

7. For [tex]\( x = 2 \)[/tex] and [tex]\( x = 3 \)[/tex]:
- [tex]\( I(2) = 55 \)[/tex] and [tex]\( I(3) = 416 \)[/tex].
- Both values are positive, no zero crossing here.

From this analysis, we find that there are zeroes:
- Between [tex]\( x = -4 \)[/tex] and [tex]\( x = -3 \)[/tex],
- Between [tex]\( x = 0 \)[/tex] and [tex]\( x = 1 \)[/tex],
- Between [tex]\( x = 1 \)[/tex] and [tex]\( x = 2 \)[/tex].

Thus, the correct description of the zeroes is:

A. Between [tex]\( x = -4 \)[/tex] and [tex]\( x = -3 \)[/tex], between [tex]\( x = 0 \)[/tex] and [tex]\( x = 1 \)[/tex], and between [tex]\( x = 1 \)[/tex] and [tex]\( x = 2 \)[/tex].