Let $$f : [0:3]\rightarrow A$$ be difined by $$f(x)=2x^{3}-15x^{2}+36x+7$$ and $$g: [0,\infty)\rightarrow B$$ be difined by $$g(x)=\frac{x^{2015}}{x^{2025}+1}$$. If both the functions are onto and $$S=\left\{x \in Z : x \in A or x \in B \right\}$$, then n(S) is equal to:

Set A (Range of $$f(x)$$):

$$f'(x) = 6x^2 - 30x + 36 = 6(x-2)(x-3)$$.

Critical points at $$x=2, 3$$.

$$f(0)=7, f(2)=35, f(3)=34$$. Range $$A = [7, 35]$$.

Set B (Range of $$g(x)$$):

$$g(0)=0$$. As $$x \to \infty, g(x) \to 0$$. Since $$g(x) \geq 0$$, the range starts at $$0$$. The max value (using AM-GM or derivative) is a very small decimal $$< 1$$. Thus, the only integer in $$B$$ is $$\{0\}$$.

Set S: Integers in $$A \cup B$$: $$\{0, 7, 8, \dots, 35\}$$.

Count: $$1$$ (for $$\{0\}$$) $$+ 29$$ (from $$7$$ to $$35$$) = 30