1. Find the codomain/range $A$ of $f$ so that $f:[0,3]\to A$ is onto

Given $f(x)=2x^3-15x^2+36x+7,\qquad x\in[0,3].$

Since the function is onto, $A$ must be the range of $f$ on $[0,3]$.

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

Critical points in $[0,3]$ are $x=2,3$.

Now check monotonicity:

• For $0\le x<2$, $f'(x)>0$ so $f$ is increasing.
• For $2<x<3$, $f'(x)<0$ so $f$ is decreasing.

So maximum occurs at $x=2$, and minimum will be at an endpoint.

Compute values: $f(0)=7,$ $f(2)=2(8)-15(4)+36(2)+7=16-60+72+7=35,$ $f(3)=2(27)-15(9)+36(3)+7=54-135+108+7=34.$

Hence the range is $A=[7,35].$

2. Find the codomain/range $B$ of $g$ so that $g:[0,\infty)\to B$ is onto

Given $g(x)=\frac{x^{2025}}{x^{2025}+1},\qquad x\in[0,\infty).$

Let $t=x^{2025}$, then $t\in[0,\infty)$ and $g(x)=\frac{t}{t+1}.$

Now:

• At $x=0$, $g(0)=0$.
• As $x\to\infty$, $g(x)\to 1$, but never equals $1$.

Thus the range is $B=[0,1).$

3. Form the set $S$

Given $S=\{x\in \mathbb Z;\ x\in A\text{ or }x\in B\}.$

This means $S$ is the set of all integers belonging to $A\cup B$.

Now:

• Integers in $A=[7,35]$ are $7,8,9,\dots,35,$ whose count is $35-7+1=29.$

• Integers in $B=[0,1)$: only $0$ belongs, since $1\notin B$.

There is no overlap between $0$ and integers from $7$ to $35$.

Hence $n(S)=29+1=30.$

4. Compare with stored answer

Derived answer is $30$, which corresponds to Option C.

So the derived answer agrees with the stored correct answer.