which function has a horizontal asymptote of y = 3?\n○ f(x)=3(2^x)\n○ f(x)=2(4)^{x - 3}\n○ f(x)=2(3^x)\n○…

Explanation:

Step1: Recall horizontal - asymptote rule for exponential functions

For an exponential function of the form $y = a(b^x)+k$, the horizontal asymptote is $y = k$ when $|b|>0$ and $b\neq1$.

Step2: Analyze each option

• For $f(x)=3(2^x)$, the form is $y = a(b^x)$ with $a = 3$ and $b = 2$. The horizontal asymptote is $y = 0$.
• For $f(x)=2(4)^{x - 3}=2\times4^{- 3}\times4^{x}=\frac{2}{64}\times4^{x}$, the form is $y = a(b^x)$ with $a=\frac{2}{64}$ and $b = 4$. The horizontal asymptote is $y = 0$.
• For $f(x)=2(3^x)$, the form is $y = a(b^x)$ with $a = 2$ and $b = 3$. The horizontal asymptote is $y = 0$.
• For $f(x)=2(4^x)+3$, the form is $y=a(b^x)+k$ where $a = 2$, $b = 4$ and $k = 3$. The horizontal asymptote is $y = 3$.

Answer:

$f(x)=2(4^x)+3$