To determine which function has a horizontal asymptote at 4, it's helpful to analyze each function individually. A horizontal asymptote represents a value that the graph of the function approaches, but never reaches, as [tex]\( x \)[/tex] approaches either positive or negative infinity.
Let’s examine each option methodically:
A. [tex]\( f(x) = -3x + 4 \)[/tex]
This function is a linear equation. Linear functions of the form [tex]\( y = mx + b \)[/tex] do not have horizontal asymptotes because their values increase or decrease without bound as [tex]\( x \)[/tex] approaches plus or minus infinity.
B. [tex]\( f(x) = 2(3)^x + 4 \)[/tex]
This function is an exponential function where the base is greater than 1. For exponential functions of the form [tex]\( f(x) = a(b)^x + c \)[/tex], the horizontal asymptote is the value [tex]\( c \)[/tex]. In this case, as [tex]\( x \)[/tex] approaches infinity, [tex]\( 2(3)^x \)[/tex] grows very large, but the function value is always shifted by 4 due to the [tex]\( +4 \)[/tex] term. Therefore, the horizontal asymptote is [tex]\( y = 4 \)[/tex].
C. [tex]\( f(x) = 3(2)^x - 4 \)[/tex]
This function is another exponential function. Here, similar to option B, the base is greater than 1. As [tex]\( x \)[/tex] approaches infinity, [tex]\( 3(2)^x \)[/tex] grows very large, but this function value is shifted by [tex]\(-4\)[/tex]. Hence, the horizontal asymptote for this function is [tex]\( y = -4 \)[/tex].
D. [tex]\( f(x) = 2x - 4 \)[/tex]
This is another linear function. Similar to the analysis of option A, linear functions do not have horizontal asymptotes.
Based on the analysis:
- Option A does not have a horizontal asymptote.
- Option B has a horizontal asymptote at [tex]\( y = 4 \)[/tex].
- Option C has a horizontal asymptote at [tex]\( y = -4 \)[/tex].
- Option D does not have a horizontal asymptote.
Thus, the function with a horizontal asymptote at 4 is:
[tex]\[ \boxed{B} \, f(x) = 2(3)^x + 4 \][/tex]
This confirms that option B is the correct answer.
