Question

shilpa’s gym membership includes a one - time fee of $20. she then pays a discounted fee of $5 for each visit. the function that shows her average cost after x visits is: f(x) = \frac{20 + 5x}{x} recall the general form of a rational function: f(x) = \frac{(a_mx^m + \dots + a_1x + a_0)}{(b_nx^n + \dots + b_1x + b_0)} which statement defines the horizontal asymptote? \boxed{m = n, so y = 0 is the horizontal asymptote.} \boxed{m = n, so y = \frac{a_m}{b_n} is the horizontal asymptote.} \boxed{m > n, so there is no horizontal asymptote.} \boxed{m < n, so y = 0 is the horizontal asymptote.}

Explanation:

Step1: Identify degrees of numerator and denominator

The function is \( f(x)=\frac{20 + 5x}{x}\). The numerator \(20+5x\) is a polynomial of degree \(m = 1\) (highest power of \(x\) is 1), and the denominator \(x\) is a polynomial of degree \(n=1\). So \(m=n\).

Step2: Recall horizontal asymptote rule for rational functions

For a rational function \(f(x)=\frac{a_mx^m+\cdots+a_1x + a_0}{b_nx^n+\cdots+b_1x + b_0}\), when \(m = n\), the horizontal asymptote is \(y=\frac{a_m}{b_n}\). In our function, \(a_m = 5\) (coefficient of \(x\) in numerator) and \(b_n=1\) (coefficient of \(x\) in denominator). So the horizontal asymptote is \(y=\frac{5}{1}=5\) which follows the rule \(m = n\), so \(y=\frac{a_m}{b_n}\) is the horizontal asymptote.

Answer:

The statement " \(m = n\), so \(y=\frac{a_m}{b_n}\) is the horizontal asymptote." (the second option among the given choices)