Question

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

Explanation:

Step1: Recall the horizontal asymptote rule for exponential functions

For an exponential function of the form \(y = a(b^{x})+c\), the horizontal asymptote is \(y = c\) when \(|b|> 0\) and \(b
eq1\).

Step2: Analyze each function

• For \(f(x)=3(2^{x})\), it is of the form \(y = a(b^{x})\) (where \(c = 0\)). The horizontal asymptote is \(y = 0\).
• For \(f(x)=2(4)^{x}-3\), it is of the form \(y=a(b^{x})+c\) with \(c=-3\). The horizontal asymptote is \(y=-3\).
• For \(f(x)=2(3^{x})\), it is of the form \(y = a(b^{x})\) (where \(c = 0\)). The horizontal asymptote is \(y = 0\).
• For \(f(x)=2(4^{x})+3\), it is of the form \(y=a(b^{x})+c\) with \(a = 2\), \(b = 4\), and \(c = 3\). Using the rule \(y = c\) for the horizontal asymptote of \(y=a(b^{x})+c\) (\(b>1\)), the horizontal asymptote is \(y = 3\).

Answer:

\(f(x)=2(4^{x})+3\) (the fourth option)