Let `f:AtoA` where `A={x:-1lexle1}`. Find whether the following function are bijective
`x|x|`

Explanation

To determine whether the function \( f: A \to A \) defined by \( f(x) = x |x| \) is bijective, we will follow these steps:

### Step 1: Define the function and the set
The function is defined as:
\[ f(x) = x |x| \]
where \( A = \{ x : -1 \leq x \leq 1 \} \).

### Step 2: Break down the function based on the value of \( x \)
We will analyze the function for two cases based on the sign of \( x \):

- **Case 1**: When \( x \geq 0 \) (i.e., \( 0 \leq x \leq 1 \)):
\[ f(x) = x \cdot x = x^2 \]

- **Case 2**: When \( x < 0 \) (i.e., \( -1 \leq x < 0 \)):
\[ f(x) = x \cdot (-x) = -x^2 \]

### Step 3: Analyze the function's behavior in each case
- For \( 0 \leq x \leq 1 \):
- The function \( f(x) = x^2 \) is a parabola that opens upwards. It ranges from \( f(0) = 0 \) to \( f(1) = 1 \).

- For \( -1 \leq x < 0 \):
- The function \( f(x) = -x^2 \) is a parabola that opens downwards. It ranges from \( f(-1) = -1 \) to \( f(0) = 0 \).

### Step 4: Graph the function
We can visualize the function:
- From \( x = -1 \) to \( x = 0 \), the graph of \( f(x) = -x^2 \) will be a downward-opening parabola.
- From \( x = 0 \) to \( x = 1 \), the graph of \( f(x) = x^2 \) will be an upward-opening parabola.

### Step 5: Determine if the function is one-to-one (injective)
To check if the function is one-to-one, we can use the horizontal line test:
- For \( x \geq 0 \), \( f(x) = x^2 \) is increasing and thus one-to-one.
- For \( x < 0 \), \( f(x) = -x^2 \) is decreasing and thus also one-to-one.

Since both parts of the function are one-to-one, the entire function is one-to-one.

### Step 6: Determine if the function is onto (surjective)
To check if the function is onto, we need to compare the range of \( f \) with its codomain \( A \):
- The range for \( x \in [0, 1] \) is \( [0, 1] \).
- The range for \( x \in [-1, 0] \) is \( [-1, 0] \).

Combining these, the overall range of \( f \) is \( [-1, 1] \), which matches the codomain \( A \).

### Conclusion
Since the function \( f(x) = x |x| \) is both one-to-one and onto, we conclude that it is bijective.