To determine what the [tex]$x$[/tex]-intercept of [tex]$f(t)$[/tex] represents, let's analyze the provided table of the height of the ball over time:
[tex]\[
\begin{tabular}{|c|c|}
\hline
Time $(t)$ & Height $f(t)$ \\
\hline
2 & 10 \\
\hline
4 & 15 \\
\hline
6 & 10 \\
\hline
8 & 0 \\
\hline
\end{tabular}
\][/tex]
The [tex]$x$[/tex]-intercept is the point where the function [tex]$f(t)$[/tex] crosses the [tex]$x$[/tex]-axis, which is when the height [tex]$f(t) = 0$[/tex]. This represents the time [tex]\( t \)[/tex] at which the ball touches the ground. From the provided table:
- At [tex]\( t = 2 \)[/tex] seconds, the height [tex]\( f(t) \)[/tex] is 10 meters.
- At [tex]\( t = 4 \)[/tex] seconds, the height [tex]\( f(t) \)[/tex] is 15 meters.
- At [tex]\( t = 6 \)[/tex] seconds, the height [tex]\( f(t) \)[/tex] is 10 meters.
- At [tex]\( t = 8 \)[/tex] seconds, the height [tex]\( f(t) \)[/tex] is 0 meters.
From this data, we observe that the ball touches the ground at [tex]\( t = 8 \)[/tex] seconds.
Therefore, the [tex]$x$[/tex]-intercept of [tex]$f(t)$[/tex] represents the time when the ball touches the ground. Hence, the correct option is:
The ball touches the ground after 8 seconds.
