To find the [tex]\( y \)[/tex]-intercept of the function [tex]\( g(x) \)[/tex] given [tex]\( g(x) = -4 f(x) + 12 \)[/tex], we must evaluate [tex]\( g(x) \)[/tex] at [tex]\( x = 0 \)[/tex].
Let’s start by analyzing the function [tex]\( f(x) \)[/tex]. The function [tex]\( f(x) = 10^2 \)[/tex] is a constant function. This means that for any value of [tex]\( x \)[/tex], [tex]\( f(x) = 100 \)[/tex].
Now, substitute [tex]\( f(x) = 100 \)[/tex] into the function [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = -4 \cdot f(x) + 12 \][/tex]
Substituting [tex]\( x = 0 \)[/tex], we get:
[tex]\[ g(0) = -4 \cdot f(0) + 12 \][/tex]
Since [tex]\( f(0) = 100 \)[/tex]:
[tex]\[ g(0) = -4 \cdot 100 + 12 \][/tex]
[tex]\[ g(0) = -400 + 12 \][/tex]
[tex]\[ g(0) = -388 \][/tex]
So, the [tex]\( y \)[/tex]-intercept of the function [tex]\( g(x) \)[/tex] is [tex]\( -388 \)[/tex]. Therefore, the coordinate of the [tex]\( y \)[/tex]-intercept is [tex]\( (0, -388) \)[/tex].
However, none of the provided answer choices match [tex]\( (0, -388) \)[/tex]. Please check if there was an error in the problem statement or the answer options, as [tex]\( (0, -388) \)[/tex] seems to be the correct [tex]\( y \)[/tex]-intercept.
