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) = (20/5x)/x. Recall the general form of a rational function: f(x) = (a_m*xᵐ + ... + a_1*x + a_0)/(b_n*xⁿ + ... + b_1*x + b_0). Which statement defines the horizontal asymptote?

1) m < n, so y = 0 is the horizontal asymptote.
2) m = n, so y = am / bn is the horizontal asymptote.
3) m = n, so y = 0 is the horizontal asymptote.
4) m > n, so there is no horizontal asymptote.

Answer 1

The horizontal asymptote for the rational function representing Shilpa's gym membership cost after x visits is y = 0 because the degree of the numerator is less than the degree of the denominator.

Step-by-step explanation:

The function that shows the average cost after x visits for Shilpa's gym membership is f(x) = (20 + 5x) / x. To identify the horizontal asymptote of a rational function, we compare the degrees of the numerator and the denominator. Since this function simplifies to f(x) = 20/x + 5, where the degree of the numerator (0, because it's a constant 20) is less than the degree of the denominator (1, for x to the first power), we follow the rule that if m < n, the horizontal asymptote is y = 0. Therefore, statement 1 is correct, and the horizontal asymptote for this function is y = 0.