To find the solution to the equation [tex]\( f(x) = g(x) \)[/tex] using three iterations of successive approximation, let's start by reviewing the functions provided:
1. [tex]\( f(x) = \frac{x^2 + 3x + 2}{x + 8} \)[/tex]
2. [tex]\( g(x) = \frac{(x - 1)}{x} \)[/tex]
Given the initial guess values from the table:
1. [tex]\(-\frac{63}{16}\)[/tex]
2. [tex]\(-\frac{61}{16}\)[/tex]
3. [tex]\(-\frac{59}{16}\)[/tex]
We'll perform iterative approximations for each step.
### Step-by-Step Solution:
#### Iteration 1
1. Initial Approximation: [tex]\( x = -\frac{63}{16} \)[/tex]
- Compute [tex]\( f(-\frac{63}{16}) \)[/tex]
- Compute [tex]\( g(-\frac{63}{16}) \)[/tex]
- Average these values to get intersection value.
2. Calculations:
[tex]\[
f\left( -\frac{63}{16} \right) = \frac{\left( -\frac{63}{16} \right)^2 + 3\left( -\frac{63}{16} \right) + 2}{\left( -\frac{63}{16} \right) + 8}
\][/tex]
[tex]\[
g\left( -\frac{63}{16} \right) = \frac{\left( -\frac{63}{16} - 1\right)}{-\frac{63}{16}}
\][/tex]
[tex]\[
\text{Intersection value} = \frac{f\left( -\frac{63}{16} \right) + g\left( -\frac{63}{16} \right)}{2}
\][/tex]
After performing the calculations, we get:
[tex]\[
\text{Intersection value} \approx 1.327464896214896
\][/tex]
#### Iteration 2
1. Next Approximation: [tex]\( x = -\frac{61}{16} \)[/tex]
- Compute [tex]\( f(-\frac{61}{16}) \)[/tex]
- Compute [tex]\( g(-\frac{61}{16}) \)[/tex]
- Average these values to get intersection value.
2. Calculations:
[tex]\[
f\left( -\frac{61}{16} \right) = \frac{\left( -\frac{61}{16} \right)^2 + 3\left( -\frac{61}{16} \right) + 2}{\left( -\frac{61}{16} \right) + 8}
\][/tex]
[tex]\[
g\left( -\frac{61}{16} \right) = \frac{\left( -\frac{61}{16} - 1\right)}{-\frac{61}{16}}
\][/tex]
[tex]\[
\text{Intersection value} = \frac{f\left( -\frac{61}{16} \right) + g\left( -\frac{61}{16} \right)}{2}
\][/tex]
After performing the calculations, we get:
[tex]\[
\text{Intersection value} \approx 1.2398229141179349
\][/tex]
#### Iteration 3
1. Next Approximation: [tex]\( x = -\frac{59}{16} \)[/tex]
- Compute [tex]\( f(-\frac{59}{16}) \)[/tex]
- Compute [tex]\( g(-\frac{59}{16}) \)[/tex]
- Average these values to get intersection value.
2. Calculations:
[tex]\[
f\left( -\frac{59}{16} \right) = \frac{\left( -\frac{59}{16} \right)^2 + 3\left( -\frac{59}{16} \right) + 2}{\left( -\frac{59}{16} \right) + 8}
\][/tex]
[tex]\[
g\left( -\frac{59}{16} \right) = \frac{\left( -\frac{59}{16} - 1\right)}{-\frac{59}{16}}
\][/tex]
[tex]\[
\text{Intersection value} = \frac{f\left( -\frac{59}{16} \right) + g\left( -\frac{59}{16} \right)}{2}
\][/tex]
After performing the calculations, we get:
[tex]\[
\text{Intersection value} \approx 1.1614084377302873
\][/tex]
### Conclusion
After three iterations of successive approximation, the approximate values for the intersection of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are:
1. 1.327464896214896
2. 1.2398229141179349
3. 1.1614084377302873
These values provide a good approximation to the solution [tex]\( x \)[/tex] where [tex]\( f(x) = g(x) \)[/tex].
