Function `f :R->R,f(x) = x|x|` is

Options

A. One - one but not onto
B. Onto but not one -one
C. One -one onto
D. Neither one - one nor onto

Answer

To determine the properties of the function \( f: \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = x |x| \), we will analyze it step by step.

### Step 1: Define the function
The function is given as:
\[ f(x) = x |x| \]
We can express the absolute value function \( |x| \) as:
\[ |x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases} \]

### Step 2: Rewrite the function based on the definition of absolute value
Using the definition of \( |x| \), we can rewrite \( f(x) \):
\[ f(x) = \begin{cases} x \cdot x = x^2 & \text{if } x \geq 0 \\ x \cdot (-x) = -x^2 & \text{if } x < 0 \end{cases} \]
Thus, we have:
\[ f(x) = \begin{cases} x^2 & \text{if } x \geq 0 \\ -x^2 & \text{if } x < 0 \end{cases} \]

### Step 3: Analyze the function for one-to-one (injective) property
To check if the function is one-to-one, we need to see if different inputs produce different outputs.

- For \( x \geq 0 \): \( f(x) = x^2 \) is a one-to-one function because if \( f(a) = f(b) \) for \( a, b \geq 0 \), then \( a^2 = b^2 \) implies \( a = b \).
- For \( x < 0 \): \( f(x) = -x^2 \) is also one-to-one because if \( f(a) = f(b) \) for \( a, b < 0 \), then \( -a^2 = -b^2 \) implies \( a^2 = b^2 \) and thus \( a = b \).

Since both cases are one-to-one, we conclude that \( f(x) \) is one-to-one.

### Step 4: Analyze the function for onto (surjective) property
To check if the function is onto, we need to see if every real number \( y \) has a corresponding \( x \) such that \( f(x) = y \).

- For \( y \geq 0 \): We can find \( x \) such that \( f(x) = y \) by solving \( x^2 = y \). The solutions are \( x = \sqrt{y} \) (for \( x \geq 0 \)).
- For \( y < 0 \): We can find \( x \) such that \( f(x) = y \) by solving \( -x^2 = y \). The solutions are \( x = -\sqrt{-y} \) (for \( x < 0 \)).

Since we can find an \( x \) for every \( y \in \mathbb{R} \), the function is onto.

### Conclusion
The function \( f(x) = x |x| \) is both one-to-one and onto. Therefore, it is a bijective function.

### Final Answer
The function \( f: \mathbb{R} \to \mathbb{R}, f(x) = x |x| \) is **one-to-one and onto**.

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