Statement-1: The function `f:R to R` defined by `f(x)=x^(3)+4x-5` is a bijection.
Statement-2: Every odd degree has at least one real root.

Options

A. 1
B. 2
C. 3
D. 4

Explanation

To determine whether the function \( f: \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = x^3 + 4x - 5 \) is a bijection, we need to show that it is both injective (one-to-one) and surjective (onto). We will also use the fact that every odd degree polynomial has at least one real root.

### Step 1: Show that \( f(x) \) is a bijection

#### 1.1: Check if \( f(x) \) is injective

To check if \( f(x) \) is injective, we need to show that if \( f(x_1) = f(x_2) \), then \( x_1 = x_2 \).

Assume \( f(x_1) = f(x_2) \):
\[ x_1^3 + 4x_1 - 5 = x_2^3 + 4x_2 - 5 \]
This simplifies to:
\[ x_1^3 - x_2^3 + 4(x_1 - x_2) = 0 \]
Factoring the left-hand side:
\[ (x_1 - x_2)(x_1^2 + x_1x_2 + x_2^2 + 4) = 0 \]
This gives us two cases:
1. \( x_1 - x_2 = 0 \) which implies \( x_1 = x_2 \).
2. \( x_1^2 + x_1x_2 + x_2^2 + 4 = 0 \).

Since \( x_1^2 + x_1x_2 + x_2^2 \) is always non-negative (as it is a sum of squares and a positive constant), the second case cannot hold. Therefore, \( f(x) \) is injective.

#### 1.2: Check if \( f(x) \) is surjective

To show that \( f(x) \) is surjective, we need to show that for every \( y \in \mathbb{R} \), there exists an \( x \in \mathbb{R} \) such that \( f(x) = y \).

Since \( f(x) \) is a cubic polynomial (odd degree), it will approach \( -\infty \) as \( x \to -\infty \) and \( +\infty \) as \( x \to +\infty \). By the Intermediate Value Theorem, since \( f(x) \) is continuous, it must take every value in \( \mathbb{R} \). Thus, \( f(x) \) is surjective.

### Conclusion

Since \( f(x) \) is both injective and surjective, we conclude that \( f(x) \) is a bijection.

### Step 2: Verify Statement 2

Statement 2 states that every odd degree polynomial has at least one real root. Since \( f(x) \) is a cubic polynomial (odd degree), it must have at least one real root, which is consistent with our findings.

### Final Conclusion

Both statements are true:
- Statement 1: The function \( f(x) = x^3 + 4x - 5 \) is a bijection.
- Statement 2: Every odd degree polynomial has at least one real root.