Question

for each function, determine whether it has a horizontal asymptote. if it does, give an equation of that horizontal asymptote.
a. ( p(x)=\frac{4 x + 8}{x - 2} )

Explanation:

Step1: Identify the degrees of numerator and denominator

For the function \( p(x)=\frac{4x + 8}{x-2} \), the degree of the numerator \(n = 1\) (since the highest - power of \(x\) in \(4x + 8\) is \(x^1\)) and the degree of the denominator \(m=1\) (since the highest - power of \(x\) in \(x - 2\) is \(x^1\)).

Step2: Use the horizontal asymptote rule for rational functions

When \(n=m\), the horizontal asymptote \(y=\frac{a_n}{b_m}\), where \(a_n\) is the leading coefficient of the numerator and \(b_m\) is the leading coefficient of the denominator.
For the numerator \(4x + 8\), \(a_n = 4\); for the denominator \(x-2\), \(b_m=1\).

Answer:

The horizontal asymptote is \(y = 4\).