To find the asymptote of the function [tex]\( f(x) = 3^x + 4 \)[/tex], we need to understand the behavior of exponential functions of the form [tex]\( f(x) = a^x + b \)[/tex].
For a general function in this form, the term [tex]\( a^x \)[/tex] grows very large as [tex]\( x \)[/tex] increases and approaches zero as [tex]\( x \)[/tex] decreases. However, the constant [tex]\( b \)[/tex] remains unaffected by the value of [tex]\( x \)[/tex].
As [tex]\( x \)[/tex] approaches negative infinity, the exponential term [tex]\( 3^x \)[/tex] approaches zero because the base 3 raised to a large negative number becomes very small. Thus, [tex]\( f(x) \)[/tex] approaches [tex]\( 4 \)[/tex].
Therefore, the horizontal asymptote of the function [tex]\( f(x) = 3^x + 4 \)[/tex] is [tex]\( y = 4 \)[/tex].
So, the asymptote is [tex]\( y = 4 \)[/tex].
