function `f: R to R: f(x) =x^(3)` is

Options

A. One One
B. Many One
C. Not a function
D. None of these

Explanation

To determine whether the function \( f: \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = x^3 \) is one-to-one and onto, we will follow these steps:

### Step 1: Show that the function is one-to-one (injective)

A function \( f \) is one-to-one if for every \( a, b \in \mathbb{R} \), \( f(a) = f(b) \) implies \( a = b \).

1. Assume \( f(a) = f(b) \).
\[ a^3 = b^3 \]
2. Taking the cube root of both sides, we have:
\[ a = b \]
3. Since \( a = b \) whenever \( f(a) = f(b) \), the function \( f(x) = x^3 \) is one-to-one.

### Step 2: Show that the function is onto (surjective)

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

1. Let \( y \) be any real number. We need to find \( x \) such that:
\[ f(x) = y \implies x^3 = y \]
2. Solving for \( x \), we find:
\[ x = y^{1/3} \]
3. Since \( y \) can be any real number, \( y^{1/3} \) is also a real number. Therefore, for every \( y \in \mathbb{R} \), there exists an \( x \in \mathbb{R} \) such that \( f(x) = y \).

### Conclusion

Since \( f(x) = x^3 \) is both one-to-one and onto, we conclude that the function \( f: \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = x^3 \) is a bijection.

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