To find the [tex]\( y \)[/tex]-intercept of the function [tex]\(\left(\frac{g}{f}\right)(x)\)[/tex], where [tex]\( f(x) = x^2 - 7x + 10 \)[/tex] and [tex]\( g(x) = x^2 - x - 20 \)[/tex], follow these steps:
1. Understand that the [tex]\( y \)[/tex]-intercept of a function is the point where the graph of the function crosses the [tex]\( y \)[/tex]-axis. This occurs when [tex]\( x = 0 \)[/tex].
2. First, substitute [tex]\( x = 0 \)[/tex] into [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] to determine their values at this point:
[tex]\[
f(0) = 0^2 - 7 \cdot 0 + 10 = 10
\][/tex]
[tex]\[
g(0) = 0^2 - 0 - 20 = -20
\][/tex]
3. Next, evaluate [tex]\(\left(\frac{g}{f}\right)(x)\)[/tex] at [tex]\( x = 0 \)[/tex], which means we need to find [tex]\(\frac{g(0)}{f(0)}\)[/tex]:
[tex]\[
\left(\frac{g}{f}\right)(0) = \frac{g(0)}{f(0)} = \frac{-20}{10} = -2
\][/tex]
4. Therefore, the [tex]\( y \)[/tex]-intercept of the function [tex]\(\left(\frac{g}{f}\right)(x)\)[/tex] is [tex]\(-2\)[/tex].
Based on this step-by-step solution, the correct answer is:
[tex]\[
\boxed{-2}
\][/tex]
