To determine which function has a horizontal asymptote of [tex]\( y = 3 \)[/tex], we need to carefully analyze each function and how they behave as [tex]\( x \)[/tex] approaches infinity ([tex]\( +\infty \)[/tex]) or negative infinity ([tex]\( -\infty \)[/tex]).
1. Function: [tex]\( f(x) = 3 \cdot 2^x \)[/tex]
As [tex]\( x \)[/tex] approaches infinity:
[tex]\[ 2^x \][/tex] grows exponentially to infinity, and so does [tex]\( 3 \cdot 2^x \)[/tex].
As [tex]\( x \)[/tex] approaches negative infinity:
[tex]\[ 2^x \][/tex] approaches 0, making [tex]\( 3 \cdot 2^x \)[/tex] also approach 0.
So, the horizontal asymptote is [tex]\( y = 0 \)[/tex].
2. Function: [tex]\( f(x) = 2 \cdot 4^{x-3} \)[/tex]
As [tex]\( x \)[/tex] approaches infinity:
Since [tex]\( 4^{x-3} \)[/tex] is an exponential function, it will grow to infinity, and so will [tex]\( 2 \cdot 4^{x-3} \)[/tex].
As [tex]\( x \)[/tex] approaches negative infinity:
[tex]\[ 4^{x-3} \][/tex] approaches 0, making [tex]\( 2 \cdot 4^{x-3} \)[/tex] also approach 0.
So, the horizontal asymptote is [tex]\( y = 0 \)[/tex].
3. Function: [tex]\( f(x) = 2 \cdot 3^x \)[/tex]
As [tex]\( x \)[/tex] approaches infinity:
[tex]\[ 3^x \][/tex] grows exponentially to infinity, and so does [tex]\( 2 \cdot 3^x \)[/tex].
As [tex]\( x \)[/tex] approaches negative infinity:
[tex]\[ 3^x \][/tex] approaches 0, making [tex]\( 2 \cdot 3^x \)[/tex] also approach 0.
So, the horizontal asymptote is [tex]\( y = 0 \)[/tex].
4. Function: [tex]\( f(x) = 2 \cdot 4^x + 3 \)[/tex]
As [tex]\( x \)[/tex] approaches infinity:
[tex]\[ 4^x \][/tex] grows exponentially to infinity, and so does [tex]\( 2 \cdot 4^x \)[/tex]. Hence, [tex]\( f(x) \)[/tex] also grows to infinity.
As [tex]\( x \)[/tex] approaches negative infinity:
[tex]\[ 4^x \][/tex] approaches 0. Thus, [tex]\( 2 \cdot 4^x \)[/tex] also approaches 0 making the function [tex]\( f(x) = 0 + 3 = 3 \)[/tex].
So, the horizontal asymptote is [tex]\( y = 3 \)[/tex].
From this analysis, we see that the function [tex]\( f(x) = 2(4^x) + 3 \)[/tex] has a horizontal asymptote of [tex]\( y = 3 \)[/tex].
Therefore, the function with a horizontal asymptote of [tex]\( y = 3 \)[/tex] is:
[tex]\[ f(x) = 2 \left(4^x\right) + 3 \][/tex]
