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We are given the function \( f(x) = x^3 + 5x + 1 \) and asked to determine its properties: whether it is one-to-one (injective), onto (surjective), both, or neither.

### 1. Injective (One-to-One) Test:
A function is injective if different inputs give different outputs, i.e., \( f(x_1) = f(x_2) \) implies \( x_1 = x_2 \). One way to test injectivity is to check the derivative of \( f(x) \):

\[
f'(x) = 3x^2 + 5
\]

Since \( 3x^2 + 5 \) is always positive for all real \( x \), \( f'(x) > 0 \). This implies that \( f(x) \) is strictly increasing. A strictly increasing function is one-to-one.

### 2. Surjective (Onto) Test:
A function is surjective if for every real number \( y \), there exists a real \( x \) such that \( f(x) = y \). In other words, the function must cover all real numbers as its range.

The function \( f(x) = x^3 + 5x + 1 \) is a cubic polynomial, and cubic polynomials have the property that their limits as \( x \to \infty \) and \( x \to -\infty \) go to \( +\infty \) and \( -\infty \), respectively. Therefore, the function will take all real values, making it surjective.

### Conclusion:
The function is both injective (one-to-one) and surjective (onto).

Thus, the correct answer is:
\[
\boxed{(c) \text{ f is one-one and onto } \mathbb{R}}
\]

Would you like further details or have any questions?

Here are five related questions to deepen understanding:
1. How can we prove that a function is neither injective nor surjective?
2. What are some other ways to test if a cubic function is injective?
3. Can a cubic function ever be neither injective nor surjective?
4. What is the significance of the derivative being always positive for injectivity?
5. How does the behavior of the function at \( \pm\infty \) influence its surjectivity?

**Tip**: A strictly increasing function is always injective since it never takes the same value twice.

For real x, let f(x) = x^3 + 5x + 1, then: (a) f is one-one but not onto R, (b) f is onto R but not one-one, (c) f is one-one and onto R, (d) f is neither one-one nor onto R.

Explore the properties of the cubic function f(x) = x^3 + 5x + 1 to determine if it is injective (one-one) and surjective (onto) over real numbers. Learn about the role of derivatives in identifying these properties, suitable for students in Grades 11-12.

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