To determine the [tex]\( x \)[/tex]-intercept of the given continuous function, we need to find the point where the function [tex]\( f(x) \)[/tex] is equal to zero. An [tex]\( x \)[/tex]-intercept occurs where the graph of the function crosses the [tex]\( x \)[/tex]-axis, i.e., [tex]\( f(x) = 0 \)[/tex].
Given the table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-2 & -10 \\
\hline
-1 & -8 \\
\hline
0 & -6 \\
\hline
1 & -4 \\
\hline
2 & -2 \\
\hline
3 & 0 \\
\hline
\end{array}
\][/tex]
We observe the [tex]\( f(x) \)[/tex] values for each corresponding [tex]\( x \)[/tex] value. We search for the point where [tex]\( f(x) = 0 \)[/tex].
Analyzing the given points:
- When [tex]\( x = -2 \)[/tex], [tex]\( f(x) = -10 \)[/tex]
- When [tex]\( x = -1 \)[/tex], [tex]\( f(x) = -8 \)[/tex]
- When [tex]\( x = 0 \)[/tex], [tex]\( f(x) = -6 \)[/tex]
- When [tex]\( x = 1 \)[/tex], [tex]\( f(x) = -4 \)[/tex]
- When [tex]\( x = 2 \)[/tex], [tex]\( f(x) = -2 \)[/tex]
- When [tex]\( x = 3 \)[/tex], [tex]\( f(x) = 0 \)[/tex]
We see that [tex]\( f(x) = 0 \)[/tex] when [tex]\( x = 3 \)[/tex]. Therefore, the point where the function [tex]\( f(x) \)[/tex] intersects the [tex]\( x \)[/tex]-axis is at [tex]\((3, 0)\)[/tex].
Hence, the [tex]\( x \)[/tex]-intercept of the continuous function is [tex]\((3, 0)\)[/tex].
