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.

Explanation:

Analyze the degrees of the numerator and denominator

We are given the rational function:

$$f(x) = \frac{25 - x^2}{x^2 - 4x - 5}$$

Let \(m\) be the degree of the numerator polynomial \(p(x) = 25 - x^2\), and \(n\) be the degree of the denominator polynomial \(q(x) = x^2 - 4x - 5\).
Using the Rational Functions concept:

• The highest power of \(x\) in the numerator is \(x^2\), so \(m = 2\).
• The highest power of \(x\) in the denominator is \(x^2\), so \(n = 2\).

Therefore, \(m = n\).

Find the horizontal asymptote

Using the Horizontal Asymptotes concept, since the degree of the numerator equals the degree of the denominator (\(m = n\)), the horizontal asymptote is given by the ratio of the leading coefficients:

• The leading coefficient of the numerator \(25 - x^2\) is \(-1\).
• The leading coefficient of the denominator \(x^2 - 4x - 5\) is \(1\).

Thus, the horizontal asymptote is:

$$y = \frac{-1}{1} = -1$$

Factor the numerator and denominator

To find the vertical asymptotes, we first factor both the numerator and the denominator to identify any common factors (which correspond to holes rather than asymptotes):

• Numerator:

$$25 - x^2 = (5 - x)(5 + x) = -(x - 5)(x + 5)$$

• Denominator:

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

We can rewrite the function as:

$$f(x) = \frac{-(x - 5)(x + 5)}{(x - 5)(x + 1)}$$

Determine the vertical asymptotes

Using the Vertical Asymptotes concept, we look at the simplified form of the rational function:

• For \(x

eq 5\), we can cancel the common factor \((x - 5)\):

$$f(x) = \frac{-(x + 5)}{x + 1}$$

• The value \(x = 5\) makes both the numerator and denominator zero, which represents a hole in the graph at \(x = 5\).
• The value \(x = -1\) makes only the denominator of the simplified fraction zero.

Thus, there is only one vertical asymptote, which is the line \(x = -1\).

Select the correct options

Based on our analysis:

• \(m = n\) is correct (since \(2 = 2\)).
• "There is only one vertical asymptote" is correct (at \(x = -1\)).
• "\(y = -1\) is the horizontal asymptote" is correct.

Answer:

• ☐ \(m

eq n\)

• ☑ \(m = n\) (Correct answer)
• ☑ There is only one vertical asymptote. (Correct answer)
• ☑ \(y = -1\) is the horizontal asymptote. (Correct answer)