The function `f:R rarr R, f(x)=x^(2)` is

Options

A. injective but not surjective
B. surjective but not injective
C. injective as well as surjective
D. neither injective nor surjective.

Answer

To determine the properties of the function \( f: \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = x^2 \), we need to check if it is injective (one-to-one) and/or surjective (onto).

### Step 1: Check if the function is injective
A function is injective if different inputs map to different outputs. In other words, if \( f(a) = f(b) \) implies that \( a = b \).

Let’s assume \( f(a) = f(b) \):
\[ a^2 = b^2 \]
This implies:
\[ a = b \quad \text{or} \quad a = -b \]
Since \( a \) can be equal to \( b \) or the negative of \( b \), we can find different inputs that yield the same output (for example, \( f(1) = 1 \) and \( f(-1) = 1 \)). Therefore, the function is **not injective**.

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

Let’s consider \( y = -4 \):
\[ f(x) = -4 \implies x^2 = -4 \]
This equation has no real solutions since the square of a real number cannot be negative. Therefore, there are values in \( \mathbb{R} \) (like -4) that are not achieved by \( f(x) \). Thus, the function is **not surjective**.

### Conclusion
Since the function \( f(x) = x^2 \) is neither injective nor surjective, the final answer is that the function is **neither injective nor surjective**.

### Final Answer
The function \( f(x) = x^2 \) is neither injective nor surjective.