To find the [tex]\( y \)[/tex]-intercept of the function [tex]\( g(x) = -4f(x) + 12 \)[/tex], we need to evaluate [tex]\( g(x) \)[/tex] at [tex]\( x = 0 \)[/tex].
First, let's recall the definition of the function [tex]\( f(x) \)[/tex]:
[tex]\[ f(x) = 10^x \][/tex]
Next, substitute [tex]\( x = 0 \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f(0) = 10^0 = 1 \][/tex]
Now, plug [tex]\( f(0) \)[/tex] into the expression for [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = -4f(x) + 12 \][/tex]
When [tex]\( x = 0 \)[/tex]:
[tex]\[ g(0) = -4f(0) + 12 \][/tex]
[tex]\[ g(0) = -4 \cdot 1 + 12 \][/tex]
[tex]\[ g(0) = -4 + 12 \][/tex]
[tex]\[ g(0) = 8 \][/tex]
Therefore, the [tex]\( y \)[/tex]-intercept of the function [tex]\( g \)[/tex] is:
[tex]\[ (0, 8) \][/tex]
Based on the given options, the correct answer is:
B. [tex]\((0, 8)\)[/tex]
