To determine which function has a horizontal asymptote at [tex]\( y = 4 \)[/tex], we need to analyze each given function and determine their behavior as [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex] or [tex]\( -\infty \)[/tex].
### Analysis of Each Function
#### A. [tex]\( f(x) = -3x + 4 \)[/tex]
This is a linear function. Linear functions do not have horizontal asymptotes because their value continuously increases or decreases without bound as [tex]\( x \)[/tex] tends to [tex]\( \infty \)[/tex] or [tex]\( -\infty \)[/tex].
#### B. [tex]\( f(x) = 3(2)^x - 4 \)[/tex]
This is an exponential function, where the base of the exponential term is positive and greater than 1. As [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex], [tex]\( 2^x \)[/tex] grows very large, and so does [tex]\( 3(2)^x \)[/tex]. Consequently, [tex]\( 3(2)^x - 4 \)[/tex] also grows without bound and does not settle at any horizontal asymptote. As [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex], [tex]\( 2^x \)[/tex] approaches 0, and thus [tex]\( 3(2)^x - 4 \)[/tex] approaches [tex]\(-4\)[/tex], indicating a horizontal asymptote at [tex]\( y = -4 \)[/tex], not [tex]\( y = 4 \)[/tex].
#### C. [tex]\( f(x) = 2x - 4 \)[/tex]
This is another linear function. Like the first function, linear functions do not have horizontal asymptotes. Their values go to [tex]\( \infty \)[/tex] or [tex]\( -\infty \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex] or [tex]\( -\infty \)[/tex], respectively.
#### D. [tex]\( f(x) = 2(3)^x + 4 \)[/tex]
This function is also exponential with a positive base greater than 1. As [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex], [tex]\( 3^x \)[/tex] grows very large, and so does [tex]\( 2(3)^x \)[/tex], causing [tex]\( 2(3)^x + 4 \)[/tex] to grow without bound. However, as [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex], [tex]\( 3^x \)[/tex] shrinks towards 0, making [tex]\( 2(3)^x \)[/tex] approach 0 as well. Thus, [tex]\( f(x) \)[/tex] approaches 4. This indicates that the function has a horizontal asymptote at [tex]\( y = 4 \)[/tex].
### Conclusion
The function with a horizontal asymptote at [tex]\( y = 4 \)[/tex] is:
[tex]\[ \boxed{D} \][/tex]
