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 \( S = \{ x \in \mathbb{Z} : x \in A { or } x \in B \} \), 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

Two functions are provided:

• \( f: [0, 3] \to A \) defined by \( f(x) = 2x^3 - 15x^2 + 36x + 7 \)
• \( g: [0, \infty) \to B \) defined by \( g(x) = \frac{x}{x^{2025} + 1} \)

Both functions are onto. The objective is to determine the cardinality of the set \( S = \{ x \in \mathbb{Z} : x \in A \text{ or } x \in B \} \).

Step 1: Analysis of Function \( f(x) \)

The function \( f(x) = 2x^3 - 15x^2 + 36x + 7 \) is a cubic polynomial. As \( f \) is onto, its range within \( x \in [0, 3] \) constitutes the set \( A \). The extreme values of \( f(x) \) for \( x \in [0, 3] \) are calculated as follows: - \( f(0) = 7 \) - \( f(3) = 34 \) Due to the continuity of \( f(x) \), the range of \( f \) over \( [0, 3] \) is \( [7, 34] \). Consequently, \( A \) comprises all integers within this interval: \[ A = \{7, 8, 9, \dots, 34\} \] The number of elements in \( A \) is: \[ n(A) = 34 - 7 + 1 = 28 \]

Step 2: Analysis of Function \( g(x) \)

The function \( g(x) = \frac{x}{x^{2025} + 1} \) is defined for \( x \geq 0 \). As \( g(x) \) is onto, its range covers the interval \( [0, 1) \) for \( x \in [0, \infty) \). The function is continuous and monotonically increasing. Therefore, the set \( B \) consists of integers within \( [0, 1) \): \[ B = \{ 0 \} \] The number of elements in \( B \) is: \[ n(B) = 1 \]

Step 3: Construction of Set \( S \)

The set \( S \) is defined as: \[ S = \{ x \in \mathbb{Z} : x \in A \text{ or } x \in B \} \] Given that \( A = \{7, 8, \dots, 34\} \) and \( B = \{0\} \), the set \( S \) includes all integers from 0 to 34: \[ S = \{ 0, 7, 8, 9, \dots, 34 \} \] The cardinality of \( S \) is calculated as: \[ n(S) = 34 - 0 + 1 = 30 \]

Final Answer:

The value of \( n(S) \) is 30.