Let `f: R->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

Explanation

To determine whether the function \( f: \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = x^4 \) is one-to-one (1-1) and onto, we will analyze the properties of the function step by step.

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

A function is one-to-one if different inputs produce different outputs. In mathematical terms, for \( f \) to be one-to-one, if \( f(a) = f(b) \), then it must follow that \( a = b \).

**Analysis**:
- Let's assume \( f(a) = f(b) \).
- This means \( a^4 = b^4 \).
- Taking the fourth root, we find \( a = b \) or \( a = -b \).

Since \( a \) can be equal to \( -b \) (for example, \( f(2) = f(-2) = 16 \)), we have found two different inputs that give the same output. Therefore, the function is **not one-to-one**.

### Step 2: Check if the function is onto

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

**Analysis**:
- The function \( f(x) = x^4 \) only produces non-negative outputs since any real number raised to an even power is non-negative.
- The range of \( f(x) \) is \( [0, \infty) \), meaning it cannot produce negative values.

Since there are values in the codomain \( \mathbb{R} \) (specifically, all negative numbers) that are not mapped to by any \( x \) in the domain, the function is **not onto**.

### Conclusion

Based on our analysis:
- The function \( f(x) = x^4 \) is **not one-to-one**.
- The function \( f(x) = x^4 \) is **not onto**.

Thus, the correct answer is that the function is neither one-to-one nor onto.

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