If `f:RtoR` be defined as `f(x)=x^(4)`. Choose the corrent answer : (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

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 determine the nature of the function \( f: \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = x^4 \), we need to analyze whether the function is one-one (injective) and onto (surjective).

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

A function is one-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, we get:
\[ |x_1| = |x_2| \]
This means \( x_1 = x_2 \) or \( x_1 = -x_2 \).

Since \( x_1 \) can be equal to \( x_2 \) or its negative, the function is not one-one.

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

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

1. The function \( f(x) = x^4 \) produces outputs that are always non-negative (i.e., \( f(x) \geq 0 \) for all \( x \)).
2. The range of \( f(x) \) is \( [0, \infty) \).
3. However, the codomain is all real numbers \( \mathbb{R} \).

Since there are no \( x \) values such that \( f(x) < 0 \), the function is not onto.

### Conclusion

Since the function \( f(x) = x^4 \) is not one-one and not onto, the correct answer is:

**(d) f is neither one-one nor onto.**

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