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

We are given two functions: 

• \( f: [0, 3] \to A \) 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, and we are asked to find the value of \( n(S) \), where \( S = \{ x \in \mathbb{Z} : x \in A \text{ or } x \in B \} \).

Step 1: Understanding the Function \( f(x) \)

The function \( f(x) = 2x^3 - 15x^2 + 36x + 7 \) is a cubic function. Since \( f \) is onto, it takes all values in the set \( A \) within the range of the function for \( x \in [0, 3] \). We need to compute the possible values of \( f(x) \) for \( x \in [0, 3] \). Evaluating \( f(x) \) at the endpoints: - \( f(0) = 2(0)^3 - 15(0)^2 + 36(0) + 7 = 7 \) - \( f(3) = 2(3)^3 - 15(3)^2 + 36(3) + 7 = 54 - 135 + 108 + 7 = 34 \) Since \( f(x) \) is continuous and onto, the range of \( f(x) \) is \( [7, 34] \), and the set \( A \) contains all integer values in this range: \[ A = \{7, 8, 9, \dots, 34\} \] Thus, the number of elements in \( A \) is: \[ n(A) = 34 - 7 + 1 = 28 \]

Step 2: Understanding the Function \( g(x) \)

The function \( g(x) = \frac{x}{x^{2025} + 1} \) is defined for \( x \geq 0 \). Since \( g(x) \) is onto, the range of \( g(x) \) spans from 0 to 1 as \( x \) increases from 0 to \( \infty \). The function \( g(x) \) is continuous and smooth, and it is strictly increasing. Hence, the set \( B \) consists of all integer values \( x \in [0, 1) \). Therefore, the set \( B \) contains only the integer 0: \[ B = \{ 0 \} \] Thus, the number of elements in \( B \) is: \[ n(B) = 1 \]

Step 3: The Set \( S \)

The set \( S \) is defined as: \[ S = \{ x \in \mathbb{Z} : x \in A \text{ or } x \in B \} \] Since \( A \) contains all integers from 7 to 34 and \( B \) contains only 0, the set \( S \) contains all integers from 0 to 34: \[ S = \{ 0, 7, 8, 9, \dots, 34 \} \] Therefore, the number of elements in \( S \) is: \[ n(S) = 34 - 0 + 1 = 30 \]

Final Answer:

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