Let `f: R to R` be defined as `f(x)=x^(4)`. Choose the correct answer

Options

A. f is one-one onto
B. f is many-one onto
C. f is one-one but not onto
D. f is neither one-one nor onto

Answer

To solve the problem, we need to analyze the function \( f: \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = x^4 \).

### Step 1: Determine 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 \( f(x_1) = f(x_2) \).
\[ x_1^4 = x_2^4 \]
2. Taking the fourth root on both sides gives:
\[ |x_1| = |x_2| \]
3. This implies that \( x_1 = x_2 \) or \( x_1 = -x_2 \).

Since there are two distinct values \( x_1 \) and \( x_2 \) (for example, \( 1 \) and \( -1 \)) that yield the same output \( f(1) = f(-1) = 1 \), the function is **not one-to-one**.

### Step 2: Determine if the function is onto (surjective)
A function is onto if for every \( y \) in the codomain \( \mathbb{R} \), there exists an \( x \) in the domain such that \( f(x) = y \).

1. The output of \( f(x) = x^4 \) is always non-negative since any real number raised to the fourth power is non-negative.
2. Therefore, the range of \( f(x) \) is \( [0, \infty) \).

Since negative values cannot be achieved by \( f(x) \), the function is **not onto**.

### Conclusion
Since the function \( f(x) = x^4 \) is neither one-to-one nor onto, we conclude that it does not satisfy the properties of a bijection.

### Final Answer
The function \( f(x) = x^4 \) is neither one-to-one nor onto.

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