Question

let ( f(x) ) be a rational function, where ( f(x) = \frac{p(x)}{q(x)} ) such that ( p(x) ) and ( q(x) ) are polynomials. suppose ( p(1) = 0 ) and ( q(1) = 9 ). then ( f(x) ) has a vertical asymptote when ( x = 1 ).
true
false
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Explanation:

Step1: Recall vertical asymptote condition

A vertical asymptote of a rational function \( f(x)=\frac{P(x)}{Q(x)} \) occurs at \( x = a \) if \( Q(a)=0 \) and \( P(a)
eq0 \) (after canceling common factors between \( P(x) \) and \( Q(x) \)).

Step2: Analyze given values

We are given \( P(1) = 0 \) and \( Q(1)=9 \). For a vertical asymptote at \( x = 1 \), we need \( Q(1)=0 \) and \( P(1)
eq0 \), but here \( Q(1)=9
eq0 \), so there is no vertical asymptote at \( x = 1 \) (instead, \( x = 1 \) is a zero of the function since \( P(1)=0 \) and \( Q(1)
eq0 \)).

Answer:

False