Let's analyze the given problem step-by-step.
1. Identify the given points and their corresponding [tex]\( g(x) \)[/tex] values:
- [tex]\( (-20, 4) \)[/tex]
- [tex]\( (-8, 0) \)[/tex]
- [tex]\( (10, 6) \)[/tex]
2. Use the relationship [tex]\( g(x) = \frac{j(x)}{x + 5} \)[/tex] to find [tex]\( j(x) \)[/tex]. As it can be re-arranged as [tex]\( j(x) = g(x) \cdot (x + 5) \)[/tex]:
- For [tex]\( x = -20 \)[/tex]:
[tex]\[
j(-20) = 4 \cdot (-20 + 5) = 4 \cdot (-15) = -60
\][/tex]
- For [tex]\( x = -8 \)[/tex]:
[tex]\[
j(-8) = 0 \cdot (-8 + 5) = 0 \cdot (-3) = 0
\][/tex]
- For [tex]\( x = 10 \)[/tex]:
[tex]\[
j(10) = 6 \cdot (10 + 5) = 6 \cdot 15 = 90
\][/tex]
3. Establish that [tex]\( j(x) \)[/tex] represents a linear function, which we denote as [tex]\( f(x) \)[/tex], thus [tex]\( f(x) = j(x) \)[/tex]. Linear functions take the form [tex]\( f(x) = mx + b \)[/tex]:
- Identify two points from the calculated [tex]\( (x, j(x)) \)[/tex] pairs:
[tex]\[
(-20, -60) \quad \text{and} \quad (-8, 0)
\][/tex]
4. Calculate the slope [tex]\( m \)[/tex] using these two points [tex]\( (x_1, y_1) = (-20, -60) \)[/tex] and [tex]\( (x_2, y_2) = (-8, 0) \)[/tex]:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - (-60)}{-8 - (-20)} = \frac{60}{12} = 5
\][/tex]
5. Determine the y-intercept [tex]\( b \)[/tex] using the point [tex]\( (-20, -60) \)[/tex] and the calculated slope [tex]\( m = 5 \)[/tex]:
[tex]\[
y - y_1 = m(x - x_1) \implies -60 = 5(-20) + b
\][/tex]
[tex]\[
-60 = -100 + b \implies b = -60 + 100 \implies b = 40
\][/tex]
6. Thus, the linear function [tex]\( f(x) \)[/tex] is:
[tex]\[
f(x) = 5x + 40
\][/tex]
7. The y-intercept, corresponding to the point [tex]\( (0, b) \)[/tex], is:
[tex]\[
(0, 40)
\][/tex]
Therefore, the y-intercept of the graph of [tex]\( y = f(x) \)[/tex] in the [tex]\( xy \)[/tex]-plane is [tex]\(\boxed{(0, 40)} \)[/tex].
