Question

consider the following function:
$f(x) = \frac{25 - x^2}{x^2 - 4x - 5}$
which of the following are correct? check all of the boxes that apply.
□ $m \
eq n$
□ $m = n$
□ there is only one vertical asymptote.
□ $y = -1$ is the horizontal asymptote.
done

Explanation:

Step1: Factor numerator and denominator

Numerator: \(25 - x^{2}=(5 + x)(5 - x)\)
Denominator: \(x^{2}-4x - 5=(x + 1)(x - 5)\)
So \(f(x)=\frac{(5 + x)(5 - x)}{(x + 1)(x - 5)}=-\frac{x + 5}{x + 1},x
eq5\)

Step2: Analyze horizontal asymptote

For a rational function \(y=\frac{a_{n}x^{n}+\cdots+a_{0}}{b_{m}x^{m}+\cdots+b_{0}}\), when \(n = m\), the horizontal asymptote is \(y=\frac{a_{n}}{b_{m}}\).
Here \(n=m = 1\), \(a_{n}=-1\), \(b_{m}=1\), so \(y=-1\)

Step3: Analyze vertical asymptote

Set the denominator of the simplified function \(x + 1=0\), \(x=-1\) (since \(x = 5\) is a removable discontinuity)

Answer:

\(m = n\), There is only one vertical asymptote, \(y=-1\) is the horizontal asymptote.