To find the approximate solution to the equation [tex]\( f(x) = g(x) \)[/tex] using three iterations of successive approximation, we will follow these steps:
### Step-by-Step Solution:
1. Define the functions:
[tex]\[
f(x) = \frac{x + 5}{x}, \quad g(x) = \frac{x - 1}{x}
\][/tex]
2. Set up the equation:
[tex]\[
\frac{x + 5}{x} = \frac{x - 1}{x}
\][/tex]
3. Combine and simplify the equation:
[tex]\[
x + 5 = x - 1
\][/tex]
This simplification is not correct in this case. Instead, graph analysis or numerical methods should be used to find the point of intersection.
4. Use the graph to approximate the initial value of [tex]\( x \)[/tex]:
From the given options, choose [tex]\(-\frac{31}{8}\)[/tex] as the initial approximation (this starting point should have ideally come from a visual inspection of the graph).
5. Formulate the iteration formula:
For the method of successive approximation:
[tex]\[
x_{n+1} = (5 \cdot x_n + x_n) / (x_n + 1)
\][/tex]
6. Set the initial approximation:
[tex]\[
x_0 = -\frac{31}{8}
\][/tex]
7. Perform three iterations:
- First Iteration:
[tex]\[
x_1 = \frac{5 \cdot x_0 + x_0}{x_0 + 1} = \frac{5 \cdot \left( -\frac{31}{8} \right) + \left( -\frac{31}{8} \right)}{\left( -\frac{31}{8} \right) + 1}
\][/tex]
- Second Iteration:
[tex]\[
x_2 = \frac{5 \cdot x_1 + x_1}{x_1 + 1}
\][/tex]
- Third Iteration:
[tex]\[
x_3 = \frac{5 \cdot x_2 + x_2}{x_2 + 1}
\][/tex]
8. Find the converging value:
After three iterations, we reach an approximate value of:
[tex]\[
x \approx 5.053584905660378
\][/tex]
So, by performing three iterations of successive approximation starting from the initial approximation [tex]\(-\frac{31}{8}\)[/tex], the solution to [tex]\( f(x) = g(x) \)[/tex] is approximately [tex]\( \boxed{5.053584905660378} \)[/tex].
