Question

Let \( f : \{1,2,3,4\} \rightarrow \{1, e, e^2, e^3\} \) is a strictly increasing and bijective function and \( g : \{1, e, e^2, e^3\} \rightarrow \{1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}\} \) strictly decreasing and bijective function. If \( \phi(x) = [f^{-1}(g^{-1}(1/2))]^x \), then find \( \int_0^1 (\phi(x) - x^2) \, dx \):

• A. \( \frac{1}{\ln 2} - \frac{1}{3} \)
• B. \( \frac{3}{\ln 2} - \frac{1}{3} \)
• C. \( \frac{2}{\ln 2} - \frac{1}{3} \)
• D. \( \frac{4}{\ln 2} - \frac{1}{3} \)

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Since \( f \) is strictly increasing and bijective, its mapping is unique: \( f(1)=1, f(2)=e, f(3)=e^2, f(4)=e^3 \). Similarly, for strictly decreasing \( g \): \( g(1)=1, g(e)=1/2, g(e^2)=1/3, g(e^3)=1/4 \).

Step 2: Key Formula or Approach:
1. Evaluate the inner function: \( g^{-1}(1/2) \). 2. Evaluate the outer function: \( f^{-1}(\dots) \). 3. Integrate the resulting exponential function.

Step 3: Detailed Explanation:
1. From \( g \), we know \( g(e) = 1/2 \), so \( g^{-1}(1/2) = e \). 2. Now find \( f^{-1}(e) \). Since \( f(2) = e \), we have \( f^{-1}(e) = 2 \). 3. Thus, \( \phi(x) = 2^x \). 4. The integral becomes: \[ \int_0^1 (2^x - x^2) \, dx = \left[ \frac{2^x}{\ln 2} - \frac{x^3}{3} \right]_0^1 \] \[ = \left( \frac{2}{\ln 2} - \frac{1}{3} \right) - \left( \frac{1}{\ln 2} - 0 \right) \] \[ = \frac{2 - 1}{\ln 2} - \frac{1}{3} = \frac{1}{\ln 2} - \frac{1}{3} \]

Step 4: Final Answer:
The result of the integral is \( \frac{1}{\ln 2} - \frac{1}{3} \).