To solve the equation [tex]\( f(x) = g(x) \)[/tex] using three iterations of successive approximation, we will iterate our initial guess through the function [tex]\( f(x) \)[/tex]. The initial guess is determined from the intersection point's approximate location on the graph.
Given the functions:
[tex]\[
f(x) = \frac{x^2 + 3x + 2}{x + 8}
\][/tex]
[tex]\[
g(x) = \frac{x - 1}{x}
\][/tex]
Starting with the initial guess [tex]\( x_0 = 1.5 \)[/tex]:
1. Calculate [tex]\( f(x_0) \)[/tex]:
[tex]\[
f(1.5) = \frac{(1.5)^2 + 3 \cdot 1.5 + 2}{1.5 + 8} = \frac{2.25 + 4.5 + 2}{9.5} = \frac{8.75}{9.5} \approx 0.921
\][/tex]
(The value is approximate for this calculation to aid understanding).
Next, use this as the new input to the function [tex]\( f \)[/tex]:
2. Calculate [tex]\( f(0.921) \)[/tex]:
[tex]\[
f(0.921) = \frac{(0.921)^2 + 3 \cdot 0.921 + 2}{0.921 + 8} = \frac{0.848 + 2.763 + 2}{8.921} = \frac{5.611}{8.921} \approx 0.629
\][/tex]
Continue the process:
3. Calculate [tex]\( f(0.629) \)[/tex]:
[tex]\[
f(0.629) = \frac{(0.629)^2 + 3 \cdot 0.629 + 2}{0.629 + 8} = \frac{0.396 + 1.887 + 2}{8.629} = \frac{4.283}{8.629} \approx 0.496
\][/tex]
After three iterations, the approximate solution converges to [tex]\( x \approx 0.496 \)[/tex]. Therefore, the approximate solution for the equation [tex]\( f(x) = g(x) \)[/tex] after three iterations of successive approximation is:
[tex]\[
x \approx 0.496
\][/tex]
