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

Options

A. one one but not on to
B. one one onto
C. onto but not one one
D. none of these

Answer

To determine the nature of the function \( f(x) = x |x| \), we will analyze whether it is one-to-one (injective) and onto (surjective).

### Step 1: Understand the function
The function is defined as:
\[ f(x) = x |x| \]
This can be rewritten based on the value of \( x \):
- If \( x \geq 0 \), then \( |x| = x \) and \( f(x) = x^2 \).
- If \( x < 0 \), then \( |x| = -x \) and \( f(x) = -x^2 \).

### Step 2: Analyze the function for \( x \geq 0 \)
For \( x \geq 0 \):
\[ f(x) = x^2 \]
The graph of \( f(x) = x^2 \) is a parabola that opens upwards. It is a one-to-one function in the interval \( [0, \infty) \) because it is strictly increasing.

### Step 3: Analyze the function for \( x < 0 \)
For \( x < 0 \):
\[ f(x) = -x^2 \]
The graph of \( f(x) = -x^2 \) is a parabola that opens downwards. It is also a one-to-one function in the interval \( (-\infty, 0) \) because it is strictly decreasing.

### Step 4: Combine the results
Since \( f(x) \) is one-to-one in both intervals \( [0, \infty) \) and \( (-\infty, 0) \), it is one-to-one overall.

### Step 5: Determine if the function is onto
Next, we check if the function is onto. The codomain of \( f(x) \) is \( \mathbb{R} \) (all real numbers).

- The range of \( f(x) \) for \( x \geq 0 \) is \( [0, \infty) \) (since \( x^2 \) is non-negative).
- The range of \( f(x) \) for \( x < 0 \) is \( (-\infty, 0] \) (since \( -x^2 \) is non-positive).

Combining these ranges, we see that the overall range of \( f(x) \) is \( (-\infty, 0] \cup [0, \infty) = \mathbb{R} \). Thus, the function is onto.

### Conclusion
Since \( f(x) \) is both one-to-one and onto, we conclude that:
\[ f(x) \text{ is one-to-one and onto.} \]

### Final Answer
The function \( f(x) = x |x| \) is **one-to-one and onto**.

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