Question

let $f(x) = \frac{p(x)}{q(x)}$ be a rational function, where p and q are polynomials in standard form with no common factors other than 1. if the degree of p equals the degree of q, which equation represents the horizontal asymptote of f?\
\\(\bigcirc\\) $y = \frac{\text{leading coefficient of p}}{\text{degree of q}}$\
\\(\bigcirc\\) $y = \text{leading coefficient of p} - \text{degree of q}$\
\\(\bigcirc\\) $y = \frac{\text{leading coefficient of p}}{\text{leading coefficient of q}}$\
\\(\bigcirc\\) $y = \text{leading coefficient of p} - \text{leading coefficient of q}$\
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question 10\
1 pts\
let $f(x) = \frac{p(x)}{q(x)}$ be a rational function, where p and q are polynomials in standard form with no common factors other than 1. if the degree of p is greater than the degree of q, how can you determine the horizontal asymptote of f, if any?\
\\(\bigcirc\\) divide the leading coefficient of p by the leading coefficient of q to find the y - value of the asymptote.\
\\(\bigcirc\\) functions of this type do not have a horizontal asymptote.\
\\(\bigcirc\\) divide the degree of p by the degree of q to find the y - value of the asymptote.\
\\(\bigcirc\\) for this case, the horizontal asymptote is always $y = 0$

Explanation:

First Question (about horizontal asymptote when degrees of p and q are equal)

Step1: Recall the rule for horizontal asymptotes of rational functions.

For a rational function \( f(x)=\frac{p(x)}{q(x)} \) where \( p \) and \( q \) are polynomials with no common factors:

• If \( \text{degree of } p=\text{degree of } q \), the horizontal asymptote is \( y = \frac{\text{leading coefficient of } p}{\text{leading coefficient of } q} \).

Step2: Analyze the options.

• Option 1: \( y=\frac{\text{leading coefficient of } p}{\text{degree of } q} \) is incorrect. The denominator should be the leading coefficient of \( q \), not the degree.
• Option 2: \( y = \text{leading coefficient of } p-\text{degree of } q \) is incorrect. The rule is about the ratio of leading coefficients, not subtraction involving degree.
• Option 3: \( y=\frac{\text{leading coefficient of } p}{\text{leading coefficient of } q} \) matches the rule.
• Option 4: \( y=\text{leading coefficient of } p - \text{leading coefficient of } q \) is incorrect. It's a ratio, not a subtraction.

Step1: Recall the rule for horizontal asymptotes of rational functions.

For a rational function \( f(x)=\frac{p(x)}{q(x)} \) where \( p \) and \( q \) are polynomials with no common factors:

• If \( \text{degree of } p>\text{degree of } q \), there is no horizontal asymptote. (If the degree of \( p \) is exactly one more than the degree of \( q \), there is an oblique asymptote, but no horizontal asymptote.)

Step2: Analyze the options.

• Option 1: Dividing leading coefficients is for when degrees are equal, so incorrect.
• Option 2: "Functions of this type do not have a horizontal asymptote" is correct.
• Option 3: Dividing degrees is not the rule for horizontal asymptotes, incorrect.
• Option 4: \( y = 0 \) is the horizontal asymptote when \( \text{degree of } p<\text{degree of } q \), so incorrect.

Answer:

\( y=\frac{\text{leading coefficient of } p}{\text{leading coefficient of } q} \) (the third option in the first question's options)

Second Question (about horizontal asymptote when degree of p is greater than degree of q)