To find the input value [tex]\(x\)[/tex] for which [tex]\(f(x)\)[/tex] is true, let’s consider the pattern in the table provided. The table enumerates the outputs [tex]\(g(x)\)[/tex] for linear functions. The values given for [tex]\(g(x)\)[/tex] are as follows:
[tex]\[
\begin{tabular}{|c|}
\hline
$g(x)$ \\
\hline
4 \\
\hline
3 \\
\hline
2 \\
\hline
1 \\
\hline
0 \\
\hline
-1 \\
\hline
\end{tabular}
\][/tex]
By examining these values, it appears that [tex]\(g(x) = x\)[/tex]. This means that for any input [tex]\(x\)[/tex], [tex]\(g(x)\)[/tex] simply returns [tex]\(x\)[/tex].
Next, we are tasked with determining for which value of [tex]\(x\)[/tex], the function [tex]\(f(x)\)[/tex] returns true. Since the instruction specifies that for [tex]\(f(x)\)[/tex] to be true, the corresponding output [tex]\(g(x)\)[/tex] in the table must match the input exactly, we look at the values to find the maximum value where this condition holds.
Analyzing the values in the table:
- [tex]\(g(4) = 4\)[/tex]
- [tex]\(g(3) = 3\)[/tex]
- [tex]\(g(2) = 2\)[/tex]
- [tex]\(g(1) = 1\)[/tex]
- [tex]\(g(0) = 0\)[/tex]
- [tex]\(g(-1) = -1\)[/tex]
For each [tex]\(x\)[/tex] provided in the table, [tex]\(g(x) = x\)[/tex].
Therefore, the largest value [tex]\(x\)[/tex] in this list (where [tex]\(g(x) = x\)[/tex]) is 4.
[tex]\[
x = 4
\][/tex]
Hence, the input value for which [tex]\(f(x) = \)[/tex] true is:
[tex]\[
\boxed{4}
\][/tex]
