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].
