Which function has a horizontal asymptote of $$ y=3 $$ ? $$ \begin{array}{l}f(x)=3\left(2^{x} ight) \ f(x)=2(4)^{x-3} \ f(x)=2\left(3^{x} ight) \ f(x)=2\left(4^{x} ight)+3\end{array} $$

Question

Which function has a horizontal asymptote of $$ y=3 $$ ? $$ \begin{array}{l}f(x)=3\left(2^{x}ight) \ f(x)=2(4)^{x-3} \ f(x)=2\left(3^{x}ight) \ f(x)=2\left(4^{x}ight)+3\end{array} $$

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We need to find the function that approaches $$ y = 3 $$ as $$ x $$ becomes very large in the negative or positive direction (depending on the function).

Consider each option:

1. $$ f(x)=3\left(2^{x}\right) $$  
   - As $$ x \to -\infty $$, $$ 2^{x} \to 0 $$ so $$ f(x) \to 0 $$.  
   - As $$ x \to \infty $$, $$ 2^{x} \to \infty $$ so $$ f(x) \to \infty $$.  
   - Neither direction gives a horizontal asymptote of $$ y=3 $$.

2. $$ f(x)=2(4)^{x-3} $$  
   - This can be rewritten as $$ f(x)=2 \cdot 4^{x} \cdot 4^{-3} $$.  
   - There is no additive constant here; the behavior is determined solely by the exponential term.  
   - As $$ x \to -\infty $$, $$ f(x) \to 0 $$; as $$ x \to \infty $$, $$ f(x) \to \infty $$.  
   - No horizontal asymptote at $$ y=3 $$ appears.

3. $$ f(x)=2\left(3^{x}\right) $$  
   - Similar reasoning: as $$ x \to -\infty $$, $$ 3^{x} \to 0 $$ so $$ f(x) \to 0 $$; as $$ x \to \infty $$, $$ f(x) \to \infty $$.  
   - No horizontal asymptote at $$ y=3 $$.

4. $$ f(x)=2\left(4^{x}\right)+3 $$  
   - Here, the function has the form "exponential function $$+$$ constant".  
   - As $$ x \to -\infty $$, $$ 4^{x} \to 0 $$ so $$ f(x) \to 2 \cdot 0 + 3 = 3 $$.  
   - This shows that the horizontal asymptote is $$ y=3 $$.

Thus, the function that has a horizontal asymptote of $$ y=3 $$ is  
$$
f(x)=2\left(4^{x}\right)+3.
$$
The function with a horizontal asymptote of $$ y=3 $$ is $$ f(x)=2\left(4^{x}\right)+3 $$.

Which function has a horizontal asymptote of ?

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