If `f:R to R ` be given by `f(x)= tan x,` then `f^(-1)(1)` is

Options

A. `(pi)/(4)`
B. `{n pi +(pi)/(4):n in Z}`
C. Does not exist
D. None of these

Explanation

To solve the problem of finding \( f^{-1}(1) \) for the function \( f(x) = \tan x \), we can follow these steps:

### Step 1: Understand the function and its inverse
Given the function \( f(x) = \tan x \), we need to find its inverse. The inverse function \( f^{-1}(y) \) is defined such that if \( y = f(x) \), then \( x = f^{-1}(y) \).

### Step 2: Set up the equation
Let \( y = \tan x \). To find the inverse, we need to express \( x \) in terms of \( y \):
\[ y = \tan x \implies x = \tan^{-1}(y) \]
Thus, we can write:
\[ f^{-1}(y) = \tan^{-1}(y) \]

### Step 3: Substitute the value
Now, we need to find \( f^{-1}(1) \):
\[ f^{-1}(1) = \tan^{-1}(1) \]

### Step 4: Evaluate \( \tan^{-1}(1) \)
We know that \( \tan(\frac{\pi}{4}) = 1 \). Therefore:
\[ \tan^{-1}(1) = \frac{\pi}{4} \]

### Step 5: Conclusion
Thus, we conclude that:
\[ f^{-1}(1) = \frac{\pi}{4} \]

### Final Answer
\[ f^{-1}(1) = \frac{\pi}{4} \]

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