To determine the approximate solutions of the system of equations where [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] intersect, we need to look for points where the values of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are close to each other.
We analyze the table:
[tex]\[
\begin{tabular}{|c|c|c|}
\hline$x$ & $f(x)$ & $g(x)$ \\
\hline 1.10 & 0.49 & 0.23 \\
\hline 1.15 & 0.32 & 0.22 \\
\hline 1.20 & 0.14 & 0.20 \\
\hline 1.25 & 0.04 & 0.18 \\
\hline 1.30 & 0.21 & 0.17 \\
\hline 1.35 & 0.39 & 0.15 \\
\hline 1.40 & 0.56 & 0.13 \\
\hline
\end{tabular}
\][/tex]
We compare [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] to find where they are almost equal:
1. At [tex]\( x = 1.10 \)[/tex]: [tex]\( f(x) = 0.49 \)[/tex] and [tex]\( g(x) = 0.23 \)[/tex]. The difference [tex]\(|0.49 - 0.23| = 0.26\)[/tex] is too large.
2. At [tex]\( x = 1.15 \)[/tex]: [tex]\( f(x) = 0.32 \)[/tex] and [tex]\( g(x) = 0.22 \)[/tex]. The difference [tex]\(|0.32 - 0.22| = 0.10\)[/tex] is still significant.
3. At [tex]\( x = 1.20 \)[/tex]: [tex]\( f(x) = 0.14 \)[/tex] and [tex]\( g(x) = 0.20 \)[/tex]. The difference [tex]\(|0.14 - 0.20| = 0.06\)[/tex] remains too large.
4. At [tex]\( x = 1.25 \)[/tex]: [tex]\( f(x) = 0.04 \)[/tex] and [tex]\( g(x) = 0.18 \)[/tex]. The difference [tex]\(|0.04 - 0.18| = 0.14\)[/tex] is also large.
5. At [tex]\( x = 1.30 \)[/tex]: [tex]\( f(x) = 0.21 \)[/tex] and [tex]\( g(x) = 0.17 \)[/tex]. The difference [tex]\(|0.21 - 0.17| = 0.04\)[/tex] is small enough to consider an intersection.
6. At [tex]\( x = 1.35 \)[/tex]: [tex]\( f(x) = 0.39 \)[/tex] and [tex]\( g(x) = 0.15 \)[/tex]. The difference [tex]\(|0.39 - 0.15| = 0.24\)[/tex] is too large.
7. At [tex]\( x = 1.40 \)[/tex]: [tex]\( f(x) = 0.56 \)[/tex] and [tex]\( g(x) = 0.13 \)[/tex]. The difference [tex]\(|0.56 - 0.13| = 0.43\)[/tex] is still too large.
From our analysis, the point [tex]\( x = 1.30 \)[/tex] has the smallest difference ([tex]\(0.04\)[/tex]), indicating that at this point, [tex]\( f(x) \approx g(x) \)[/tex], and thus there is an approximate intersection.
Therefore, the best approximate solutions of the system of equations are:
- The solutions exist where [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] intersect at [tex]\( x = 1.30 \)[/tex].
