Step 1 :The problem is asking for the correct representation of the cost of purchasing \$360 worth of goods when the '40\% off' coupon is applied first followed by the '\$15 off' coupon.
Step 2 :The '40\% off' coupon is represented by the function g and the '\$15 off' coupon is represented by the function f.
Step 3 :Since the '40\% off' coupon is applied first, we first input the original cost into the function g. Then, the output of g (which is the cost after the '40\% off' coupon is applied) is inputted into the function f (which represents the '\$15 off' coupon).
Step 4 :Therefore, the correct representation is \(f(g(360))=201\).
Step 5 :Final Answer: \(\boxed{f(g(360))=201}\)