Question

1. what is the horizontal asymptote of the function $f(x) = 3(2)^x + 7$?

a. $y = 2$
b. $y = 3$
c. $y = 7$
d. $y = 0$

Explanation:

Step1: Recall exponential function asymptote

For an exponential function of the form \( f(x) = a(b)^x + k \), the horizontal asymptote is determined by the constant term \( k \). As \( x \to -\infty \) (for \( b > 1 \)), \( b^x \to 0 \).

Step2: Analyze the given function

The function is \( f(x)=3(2)^x + 7 \). Here, \( a = 3 \), \( b = 2 \) (which is \( > 1 \)), and \( k = 7 \). When \( x \to -\infty \), \( 2^x \to 0 \), so \( 3(2)^x \to 0 \). Then \( f(x) \to 0 + 7 = 7 \). So the horizontal asymptote is \( y = 7 \).

Answer:

C. \( y = 7 \)