Question

question
which of the following are the vertical asymptotes of the graph of ( f ( x ) = \frac { x ^ { 2 } - 25 } { x ^ { 2 } - 5 x + 6 } )?
select all that apply.
select all that apply.
vertical asymptote at ( x = 5 ).
vertical asymptote at ( x = - 5 ).
vertical asymptote at ( x = 3 ).
vertical asymptote at ( x = 2 ).
there are no vertical asymptotes.

Explanation:

Step1: Factor numerator and denominator

Factor \(x^{2}-25=(x + 5)(x - 5)\) and \(x^{2}-5x + 6=(x-2)(x - 3)\). So \(f(x)=\frac{(x + 5)(x - 5)}{(x-2)(x - 3)}\).

Step2: Find vertical asymptotes

Vertical asymptotes occur where the denominator is zero (and numerator is non - zero). Set denominator \((x-2)(x - 3)=0\), we get \(x = 2\) and \(x=3\). When \(x = 2\), numerator \((2 + 5)(2 - 5)=-21
eq0\). When \(x = 3\), numerator \((3 + 5)(3 - 5)=-16
eq0\).

Answer:

Vertical asymptote at \(x = 2\), Vertical asymptote at \(x = 3\)