To find the [tex]\( y \)[/tex]-intercept of the function [tex]\( g \)[/tex] given by [tex]\( g(x) = -4 f(x) + 12 \)[/tex], follow these steps:
1. Understand the concept of the [tex]\( y \)[/tex]-intercept:
The [tex]\( y \)[/tex]-intercept of a function is the point where the graph of the function intersects the [tex]\( y \)[/tex]-axis. This occurs when [tex]\( x = 0 \)[/tex].
2. Substitute [tex]\( x = 0 \)[/tex] into the function:
We need to evaluate [tex]\( g(0) \)[/tex] by substituting [tex]\( x = 0 \)[/tex] into the given function [tex]\( g(x) \)[/tex].
3. Calculate [tex]\( g(0) \)[/tex]:
[tex]\[
g(0) = -4 f(0) + 12
\][/tex]
4. Focus on the constant term:
In the expression [tex]\( -4 f(0) + 12 \)[/tex], the value of [tex]\( f(0) \)[/tex] is not provided, so let's focus on the constant term which is [tex]\( 12 \)[/tex].
5. Confirm the contribution of [tex]\( f(0) \)[/tex]:
Since [tex]\( g(0) \)[/tex] results in [tex]\( -4 f(0) + 12 \)[/tex] and the absence of specifics about [tex]\( f(0) \)[/tex] does not affect the intercept calculation, we rely on the constant term [tex]\( +12 \)[/tex].
6. Identify the [tex]\( y \)[/tex]-intercept:
After evaluating [tex]\( g(0) \)[/tex], the [tex]\( y \)[/tex]-intercept is given by [tex]\((0, 12)\)[/tex].
Thus, the [tex]\( y \)[/tex]-intercept of the function [tex]\( g \)[/tex] is [tex]\((0, 12)\)[/tex].
So, the correct answer is:
C. [tex]\((0, 12)\)[/tex]
