Let's approximate the solution to the equation [tex]\( f(x) = g(x) \)[/tex] using three iterations of successive approximation.
Given:
[tex]\[
f(x) = \frac{x^2 + 2x + 2}{x + 8}
\][/tex]
[tex]\[
g(x) = \frac{x - 1}{x}
\][/tex]
We'll start the successive approximation process with an initial guess for [tex]\( x \)[/tex] based on the graph. Let's assume our initial guess is [tex]\( x_0 = 1.5 \)[/tex].
### Iteration 1
1. Calculate [tex]\( f(x_0) \)[/tex]:
[tex]\[
f(1.5) = \frac{(1.5)^2 + 2(1.5) + 2}{1.5 + 8} = \frac{2.25 + 3 + 2}{9.5} = \frac{7.25}{9.5} \approx 0.763
\][/tex]
2. Calculate [tex]\( g(x_0) \)[/tex]:
[tex]\[
g(1.5) = \frac{1.5 - 1}{1.5} = \frac{0.5}{1.5} \approx 0.333
\][/tex]
3. Compute the new [tex]\( x_1 \)[/tex] using the average of [tex]\( f(x_0) \)[/tex] and [tex]\( g(x_0) \)[/tex]:
[tex]\[
x_1 = \frac{f(x_0) + g(x_0)}{2} = \frac{0.763 + 0.333}{2} \approx 0.548
\][/tex]
### Iteration 2
1. Update [tex]\( x_0 = x_1 \)[/tex]. Now, [tex]\( x_0 \approx 0.548 \)[/tex].
2. Calculate [tex]\( f(x_0) \)[/tex]:
[tex]\[
f(0.548) = \frac{(0.548)^2 + 2(0.548) + 2}{0.548 + 8} = \frac{0.300304 + 1.096 + 2}{8.548} = \frac{3.396304}{8.548} \approx 0.397
\][/tex]
3. Calculate [tex]\( g(x_0) \)[/tex]:
[tex]\[
g(0.548) = \frac{0.548 - 1}{0.548} = \frac{-0.452}{0.548} \approx -0.825
\][/tex]
4. Compute the new [tex]\( x_1 \)[/tex] using the average of [tex]\( f(x_0) \)[/tex] and [tex]\( g(x_0) \)[/tex]:
[tex]\[
x_1 = \frac{f(x_0) + g(x_0)}{2} = \frac{0.397 + (-0.825)}{2} \approx -0.214
\][/tex]
### Iteration 3
1. Update [tex]\( x_0 = x_1 \)[/tex]. Now, [tex]\( x_0 \approx -0.214 \)[/tex].
2. Calculate [tex]\( f(x_0) \)[/tex]:
[tex]\[
f(-0.214) = \frac{(-0.214)^2 + 2(-0.214) + 2}{-0.214 + 8} = \frac{0.045796 + (-0.428) + 2}{7.786} = \frac{1.617796}{7.786} \approx 0.208
\][/tex]
3. Calculate [tex]\( g(x_0) \)[/tex]:
[tex]\[
g(-0.214) = \frac{-0.214 - 1}{-0.214} = \frac{-1.214}{-0.214} \approx 5.673
\][/tex]
4. Compute the new [tex]\( x_1 \)[/tex] using the average of [tex]\( f(x_0) \)[/tex] and [tex]\( g(x_0) \)[/tex]:
[tex]\[
x_1 = \frac{f(x_0) + g(x_0)}{2} = \frac{0.208 + 5.673}{2} \approx 2.941
\][/tex]
After three iterations, the successive approximation process yields an approximate solution of [tex]\( x \approx 2.948 \)[/tex].
Therefore, the approximate value of the solution to the equation [tex]\( f(x) = g(x) \)[/tex] after three iterations is:
[tex]\[
\boxed{2.948}
\][/tex]
