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: $x^2-4x-5=(x-5)(x+1)=-(5-x)(x+1)$

Step2: Simplify the function

Cancel common factor $(5-x)$ (for $x
eq5$):
$f(x)=\frac{(5-x)(5+x)}{-(5-x)(x+1)}=\frac{5+x}{-(x+1)}=-\frac{x+5}{x+1}$

Step3: Find vertical asymptotes

Vertical asymptotes occur where denominator is 0 (after simplifying, excluding holes):
$x+1=0 \implies x=-1$
The value $x=5$ is a hole, not an asymptote, so there is 1 vertical asymptote.

Step4: Find horizontal asymptote

For rational functions, compare degrees of numerator and denominator. Both are degree 1. Horizontal asymptote is ratio of leading coefficients:
$y=\frac{1}{-1}=-1$

Step5: Analyze $m$ and $n$ (degrees)

Degree of numerator $m=2$, degree of denominator $n=2$, so $m=n$.

Answer:

• $m = n$
• There is only one vertical asymptote.
• $y = -1$ is the horizontal asymptote.