Final answer:
Function f for the athletic club is f(x) = 15x + 25 with a slope of 15 and a y-intercept of 25. Function g, where the application fee is waived, is g(x) = 15x with the same slope of 15 and a y-intercept of 0. Both functions have the same rate of increase per month, but they differ in their initial starting costs due to the application fee.
Step-by-step explanation:
To write the functions for the athletic club's membership fees, we'll use linear equations in the form of y = mx + b, where m is the slope and b is the y-intercept. For function f, which models the total cost including the application fee, the equation is f(x) = 15x + 25. The slope is 15, representing the monthly membership fee of $15 per month, and the y-intercept is 25, representing the one-time application fee of $25.
Function g represents the cost of membership if the application fee is waived, so the equation is g(x) = 15x. The slope here is also 15 (which is the same as the slope of function f), while the y-intercept is 0, as there's no application fee.
The slopes of both functions are the same, indicating that the cost of membership increases at the same rate per month for both. However, the y-intercepts differ; f(x) starts at $25 due to the application fee, while g(x) starts at $0.