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What is the approximate value for the solution to the equation [tex]$f(x) = g(x)$[/tex]?

\begin{tabular}{|c|c|}
\hline
\multicolumn{2}{|c|}{
\begin{tabular}{c}
Successive \\
Approximation \\
Intersection Values
\end{tabular}} \\
\hline
[tex]$\frac{5}{8}$[/tex] & [tex]$\frac{11}{16}$[/tex] \\
\hline
[tex]$\frac{3}{4}$[/tex] & [tex]$\frac{13}{16}$[/tex] \\
\hline
[tex]$\frac{7}{8}$[/tex] & [tex]$\frac{15}{16}$[/tex] \\
\hline
\end{tabular}

Sagot :

GinnyAnswer
To approximate the solution to the equation [tex]\( f(x) = g(x) \)[/tex] using the given successive approximation intersection values, we can follow these steps:

1. Understand the Values:
We are given a table of fractions that appear to represent points where the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] intersect. These fractions are given in pairs as follows:
- First pair: [tex]\( \frac{5}{8} \)[/tex] and [tex]\( \frac{11}{16} \)[/tex]
- Second pair: [tex]\( \frac{3}{4} \)[/tex] and [tex]\( \frac{13}{16} \)[/tex]
- Third pair: [tex]\( \frac{7}{8} \)[/tex] and [tex]\( \frac{15}{16} \)[/tex]

2. Convert Fractions to Decimals:
Convert each fraction to its decimal equivalent for easier comparison:
- [tex]\( \frac{5}{8} = 0.625 \)[/tex]
- [tex]\( \frac{11}{16} = 0.6875 \)[/tex]
- [tex]\( \frac{3}{4} = 0.75 \)[/tex]
- [tex]\( \frac{13}{16} = 0.8125 \)[/tex]
- [tex]\( \frac{7}{8} = 0.875 \)[/tex]
- [tex]\( \frac{15}{16} = 0.9375 \)[/tex]

3. Analyze the Values:
We see that these decimal values increase successively, indicating closer approximations to the intersection point.

4. Determine the Approximate Solution:
Generally, a good approximation can be found by averaging the middle successive values. Here, those would be [tex]\( \frac{3}{4} = 0.75 \)[/tex] and [tex]\( \frac{13}{16} = 0.8125 \)[/tex].

Calculate the average of these two middle values:
[tex]\[ \frac{0.75 + 0.8125}{2} = 0.78125 \][/tex]

5. Conclusion:
Based on our analysis and averaging the two middle values, the approximate solution to the equation [tex]\( f(x) = g(x) \)[/tex] is:
[tex]\[ x \approx 0.78125 \][/tex]

So, the approximate value for the solution is [tex]\( 0.78125 \)[/tex].
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