If `f:R to R` is defined by` f(x)=|x|,` then

Options

A. `f ^(-1)(x) =-x`
B. `f ^(-1) (x) = (1)/(|x|)`
C. The function `f ^(-1) (x)` does not exist
D. `f ^(-1)(x) =1/x`

Answer

To solve the problem, we need to analyze the function \( f: \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = |x| \) and determine whether it has an inverse.

### Step 1: Understanding the Function
The function \( f(x) = |x| \) takes any real number \( x \) and returns its absolute value. This means:
- For \( x \geq 0 \), \( f(x) = x \).
- For \( x < 0 \), \( f(x) = -x \).

### Step 2: Checking if the Function is One-One
A function is one-one (injective) if different inputs produce different outputs. To check if \( f(x) \) is one-one, we can consider:
- \( f(a) = f(b) \) implies \( |a| = |b| \).
This means that \( a \) could be equal to \( b \) or \( a \) could be equal to \( -b \). Thus, two different inputs can yield the same output (for example, \( f(2) = f(-2) = 2 \)). Therefore, \( f(x) \) is not one-one.

### Step 3: Checking if the Function is Onto
A function is onto (surjective) if every element in the codomain (which is \( \mathbb{R} \) in this case) is mapped by some element in the domain. The range of \( f(x) = |x| \) is \( [0, \infty) \), which does not cover all real numbers (e.g., negative numbers are not included). Therefore, \( f(x) \) is not onto.

### Step 4: Conclusion about the Inverse
Since the function \( f(x) = |x| \) is neither one-one nor onto, it does not have an inverse. Thus, we conclude that \( f^{-1}(x) \) does not exist.

### Final Answer
The inverse function \( f^{-1}(x) \) does not exist.

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