Question

select the correct answer.
when graphed, which function has a horizontal asymptote at 4?
a. ( f(x)=2(3)^{x}+4 )
b. ( f(x)=2x - 4 )
c. ( f(x)=3(2)^{x}-4 )
d. ( f(x)=-3x + 4 )

Explanation:

Step1: Recall the horizontal asymptote rules

For an exponential function of the form \(y = a(b)^{x}+k\), the horizontal asymptote is \(y = k\). For a linear function \(y=mx + n\), there is no horizontal asymptote (it is a straight - line).

Step2: Analyze each option

• Option A:

For the function \(f(x)=2(3)^{x}+4\), since it is in the form \(y = a(b)^{x}+k\) with \(a = 2\), \(b=3\), and \(k = 4\). The horizontal asymptote is \(y = 4\).

• Option B:

The function \(f(x)=2x - 4\) is a linear function (\(y=mx + n\) where \(m = 2\) and \(n=-4\)). Linear functions do not have horizontal asymptotes.

• Option C:

For the function \(f(x)=3(2)^{x}-4\), it is in the form \(y=a(b)^{x}+k\) with \(a = 3\), \(b = 2\), and \(k=-4\). The horizontal asymptote is \(y=-4\).

• Option D:

The function \(f(x)=-3x + 4\) is a linear function (\(y=mx + n\) where \(m=-3\) and \(n = 4\)). Linear functions do not have horizontal asymptotes.

Answer:

A. \(f(x)=2(3)^{x}+4\)