Let f be an injective map with domain {x,y,z) and range {1,2,3} such that exactly one of the following statements is correct and the remaining are false : `f (x) = 1, f (y) ne 1,f (z) ne 2.` The value of `f^(-1)` (1) is

Options

A. x
B. y
C. z
D. none of these

Answer

To solve the problem, we need to analyze the statements given about the injective function \( f \) with domain \{x, y, z\} and range \{1, 2, 3\}. We know that exactly one of the following statements is true:

1. \( f(x) = 1 \)
2. \( f(y) \neq 1 \)
3. \( f(z) \neq 2 \)

Since \( f \) is an injective function, each element in the domain must map to a unique element in the range.

### Step 1: Analyze the statements

Let's analyze each statement to determine which one can be true.

- **Assume Statement 1 is true**: \( f(x) = 1 \)
- If \( f(x) = 1 \), then \( f(y) \) and \( f(z) \) must map to the remaining values in the range, which are 2 and 3.
- Thus, \( f(y) \) can be either 2 or 3, and \( f(z) \) will take the remaining value.
- This means \( f(y) \neq 1 \) is true, and \( f(z) \neq 2 \) can be either true or false depending on the assignment of values. Therefore, this case violates the condition that only one statement is true.

- **Assume Statement 2 is true**: \( f(y) \neq 1 \)
- If \( f(y) \neq 1 \), then \( f(y) \) must be either 2 or 3.
- This means \( f(x) \) can either be 1 or 2, but if \( f(x) = 1 \), then \( f(z) \) must be 3, which means \( f(z) \neq 2 \) is true as well. This again violates the condition that only one statement is true.

- **Assume Statement 3 is true**: \( f(z) \neq 2 \)
- If \( f(z) \neq 2 \), then \( f(z) \) can either be 1 or 3.
- If \( f(z) = 1 \), then \( f(x) \) must be 2, and \( f(y) \) must be 3, which means \( f(y) \neq 1 \) is true, and this case holds.
- If \( f(z) = 3 \), then \( f(x) \) can be 1 or 2, and \( f(y) \) must take the remaining value. In this case, \( f(y) \neq 1 \) can still hold true.

Thus, the only consistent scenario is when Statement 3 is true.

### Step 2: Determine the values of \( f \)

From our analysis, we can conclude:
- \( f(z) \) must be 3 (since \( f(z) \neq 2 \)).
- This leaves us with \( f(x) \) and \( f(y) \) to take the values 1 and 2.
- We can assign \( f(x) = 1 \) and \( f(y) = 2 \).

### Step 3: Find \( f^{-1}(1) \)

Since we have determined that \( f(x) = 1 \), we can find the inverse function value:
- \( f^{-1}(1) = x \)

### Final Answer

The value of \( f^{-1}(1) \) is \( x \).

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