Question

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

• A. 30
• B. 36
• C. 29
• D. 31

Hint:

When dealing with ranges and functions, always consider the behavior of the function and its derivative to understand its range.

Answer 1

The Correct Option is A

Given that \( f(x) \) is surjective, its range is \( A \).
The derivative is \( f'(x) = 6x^2 - 30x + 36 \), which factors to \( f'(x) = 6(x-2)(x-3) \). 
Evaluating \( f(x) \) at key points yields: \( f(2) = 35 \), \( f(3) = 34 \), and \( f(0) = 7 \). 
Consequently, the range of \( f(x) \) is \( [7, 35] \). 
For \( g(x) = \frac{1}{x^{2025} + 1} \), the range is \( [0, 1] \). 
Therefore, the set \( S = \{ 0, 7, 8, \ldots, 35 \} \) contains 30 elements.