To approximate the solution to the equation [tex]\( f(x) = g(x) \)[/tex] using successive approximation, we first need to understand both functions.
Given:
[tex]\[ f(x) = \frac{x^2 + x + 2}{x + 1} \][/tex]
[tex]\[ g(x) = \frac{x - 1}{x} \][/tex]
We need to find the value of [tex]\( x \)[/tex] where these functions are equal:
[tex]\[ \frac{x^2 + x + 2}{x + 1} = \frac{x - 1}{x} \][/tex]
By analyzing and solving the equation step-by-step, we obtained the approximate values for the solution as:
[tex]\[ 0.226698825758202 \pm 1.46771150871022i \][/tex]
[tex]\[ -0.453397651516404 \][/tex]
From the given table, the approximation values are in fraction forms. Let's convert our complex solutions into a meaningful approximate fractional form in context with the table.
Upon comparing our approximate solutions with the values in the table we notice that:
[tex]\[ -0.453397651516404 \approx -\frac{24}{16} \][/tex]
This matching approximate solution is critical in our identification step. Therefore, the third row of the table [tex]\(-\frac{24}{16}\)[/tex] represents one of our approximate intersection values.
Hence the correct location in the table is where [tex]\(-\frac{24}{16}\)[/tex] is listed, which is in the third row. The final correct answer is:
\begin{tabular}{|l|l|}
\hline
Successive Approximation Intersection Values \\
\hline
-\frac{24}{16} \\
\hline
\end{tabular}
Thus, you should mark or select the entry [tex]\(\boxed{-\frac{24}{16}}\)[/tex].
