Given `f(x) = tan(sqrt(pi^2/16 - x^2))` and A = R - [0,1]

Options

A. Range of f (x) is A
B. Range of f(x) is A'
C. Maximum value of f(x)=1
D. Minimum value of `(f(x))^(-1)=1`

Answer

To solve the given problem, we need to analyze the function \( f(x) = \tan\left(\sqrt{\frac{\pi^2}{16} - x^2}\right) \) and determine its range when \( A = \mathbb{R} - [0, 1] \).

### Step 1: Identify the Domain of the Function
The expression inside the square root, \( \frac{\pi^2}{16} - x^2 \), must be non-negative for \( f(x) \) to be real. Therefore, we set up the inequality:
\[ \frac{\pi^2}{16} - x^2 \geq 0 \]
This simplifies to:
\[ x^2 \leq \frac{\pi^2}{16} \]
Taking the square root of both sides gives:
\[ |x| \leq \frac{\pi}{4} \]
Thus, the domain of \( f(x) \) is:
\[ x \in \left[-\frac{\pi}{4}, \frac{\pi}{4}\right] \]

### Step 2: Find the Maximum and Minimum Values of \( f(x) \)
To find the maximum and minimum values of \( f(x) \), we evaluate \( f(x) \) at the endpoints of the domain and at \( x = 0 \):
1. At \( x = 0 \):
\[ f(0) = \tan\left(\sqrt{\frac{\pi^2}{16} - 0^2}\right) = \tan\left(\frac{\pi}{4}\right) = 1 \]

2. At \( x = \frac{\pi}{4} \):
\[ f\left(\frac{\pi}{4}\right) = \tan\left(\sqrt{\frac{\pi^2}{16} - \left(\frac{\pi}{4}\right)^2}\right) = \tan(0) = 0 \]

3. At \( x = -\frac{\pi}{4} \):
\[ f\left(-\frac{\pi}{4}\right) = \tan\left(\sqrt{\frac{\pi^2}{16} - \left(-\frac{\pi}{4}\right)^2}\right) = \tan(0) = 0 \]

From these evaluations, we find that:
- The maximum value of \( f(x) \) is \( 1 \) (at \( x = 0 \)).
- The minimum value of \( f(x) \) is \( 0 \) (at \( x = \pm\frac{\pi}{4} \)).

### Step 3: Determine the Range of \( f(x) \)
Since \( f(x) \) takes values from \( 0 \) to \( 1 \), the range of \( f(x) \) is:
\[ [0, 1] \]

### Step 4: Identify the Complement of the Range
Given that \( A = \mathbb{R} - [0, 1] \), the complement of the range of \( f(x) \) is:
\[ A = (-\infty, 0) \cup (1, \infty) \]

### Conclusion
Thus, the correct options based on the analysis are:
- The maximum value of \( f(x) \) is \( 1 \).
- The minimum value of \( f(x) \) is \( 0 \).