To identify the [tex]\( y \)[/tex]-intercepts of the continuous function given the table, we look for all points where the function value [tex]\( f(x) \)[/tex] is equal to zero. The [tex]\( y \)[/tex]-intercept of a function is where the function crosses the [tex]\( y \)[/tex]-axis, which happens when the [tex]\( y \)[/tex]-coordinate (or [tex]\( f(x) \)[/tex]) is zero.
Here's a step-by-step process:
1. Check all the [tex]\( x \)[/tex] values in the table and their corresponding [tex]\( f(x) \)[/tex] values:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-2 & -8 \\
\hline
-1 & 0 \\
\hline
0 & 0 \\
\hline
1 & -2 \\
\hline
2 & 0 \\
\hline
3 & 12 \\
\hline
\end{array}
\][/tex]
2. Identify the points in the table where [tex]\( f(x) = 0 \)[/tex]:
- For [tex]\( x = -1 \)[/tex], [tex]\( f(x) = 0 \)[/tex]
- For [tex]\( x = 0 \)[/tex], [tex]\( f(x) = 0 \)[/tex]
- For [tex]\( x = 2 \)[/tex], [tex]\( f(x) = 0 \)[/tex]
3. List the points where [tex]\( f(x) = 0 \)[/tex]. These are [tex]\( y \)[/tex]-intercepts:
- [tex]\( (-1, 0) \)[/tex]
- [tex]\( (0, 0) \)[/tex]
- [tex]\( (2, 0) \)[/tex]
Hence, the correct list of all [tex]\( y \)[/tex]-intercepts of the continuous function in the table is [tex]\( (-1,0), (0,0), (2,0) \)[/tex].
The correct answer is:
[tex]\( (-1,0), (0,0), (2,0) \)[/tex]
