Let `f:AtoA` where `A={x:-1lexle1}`. Find whether the following function are bijective
`x^(4)`

Explanation

To determine whether the function \( f: A \to A \) defined by \( f(x) = x^4 \) is bijective, we need to check if it is both one-to-one (injective) and onto (surjective).

### Step 1: Check if the function is one-to-one (injective)

A function is one-to-one if \( f(x_1) = f(x_2) \) implies \( x_1 = x_2 \).

1. Assume \( f(x_1) = f(x_2) \).
2. This means \( x_1^4 = x_2^4 \).
3. Taking the fourth root, we get \( |x_1| = |x_2| \).
4. This implies \( x_1 = x_2 \) or \( x_1 = -x_2 \).

Since there exist distinct \( x_1 \) and \( x_2 \) (for example, \( x_1 = 1 \) and \( x_2 = -1 \)) such that \( f(x_1) = f(x_2) \), we conclude that \( f \) is not one-to-one.

### Step 2: Check if the function is onto (surjective)

A function is onto if for every element \( y \) in the codomain \( A \), there exists an \( x \) in the domain \( A \) such that \( f(x) = y \).

1. The codomain \( A = \{ x : -1 \leq x \leq 1 \} \).
2. The range of \( f(x) = x^4 \) for \( x \in A \) is \( [0, 1] \) because:
- The minimum value occurs at \( x = 0 \) (where \( f(0) = 0^4 = 0 \)).
- The maximum value occurs at \( x = 1 \) or \( x = -1 \) (where \( f(1) = 1^4 = 1 \)).
3. The range of \( f \) is \( [0, 1] \), which does not cover the entire codomain \( A \) since \( A \) includes negative numbers (e.g., \( -1 \)).
4. Therefore, there are elements in \( A \) (like \( -0.5 \)) for which there is no \( x \in A \) such that \( f(x) = -0.5 \).

Since the function is not onto, we conclude that \( f \) is not bijective.

### Conclusion

The function \( f(x) = x^4 \) is neither one-to-one nor onto, and therefore it is not bijective.

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