For real x, let f(x) = x³ + 5x + 1, then (a) f is onto R but not one-one (b) f is one-one and onto R

Q: Wait, why do we reject the negative root here? Or do we even have roots?

Great question! We don't actually need to find the roots of $f(x) = 0$ to determine if it is one-one. The derivative test tells us about the *slope* of the function. Since $f'(x) = 3x^2 + 5$ is always positive, the function is strictly increasing. A strictly increasing function can have at most one real root, but it never 'turns back' to hit the same $y$-value twice, which is the definition of one-one.

Q: Why can't we just use energy conservation or something else?

Energy conservation is a physical principle, but here we are dealing with pure calculus and function analysis. The 'tool' for analyzing function behavior (one-one and onto) is the derivative test and limit analysis. Stick to the mathematical definitions: one-one means $f(x_1) = f(x_2) \implies x_1 = x_2$, and onto means the range equals the codomain.

Q: Why does the range of a cubic polynomial always cover all real numbers?

Think about the limits. As $x \to \infty$, $x^3$ dominates, so $f(x) \to \infty$. As $x \to -\infty$, $x^3$ also dominates, so $f(x) \to -\infty$. Because polynomials are continuous, the Intermediate Value Theorem guarantees that the function must take on every value between $-\infty$ and $\infty$. Thus, the range is indeed all real numbers.

Question

For real $x$, let $f(x) = x^3 + 5x + 1$, then

Options:

(A) $f$ is onto $R$ but not one-one

(B) $f$ is one-one and onto $R$

(C) $f$ is neither one-one nor onto $R$

(D) $f$ is one-one but not onto $R$.

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing on a vast, infinite plain, looking at the graph of a function. The function is defined as $f(x) = x^3 + 5x + 1$.

Our mission is to determine its identity: is it one-one? Is it onto? These are the two fundamental questions that define the soul of a function.

To be one-one, or injective, means that no two different inputs can produce the same output. It is a 'no-collision' rule.

To be onto, or surjective, means that every possible output in the codomain is reached by at least one input. It is a 'no-gap' rule.

### The One-One Mystery: The Power of the Derivative

How do we know if a function is one-one without drawing it? We use the most powerful tool in our calculus arsenal: the derivative.

If a function is strictly increasing or strictly decreasing, it can never return to a previous $y$-value. It is always moving forward.

Let us calculate the derivative of our function:
$$f'(x) = \frac{d}{dx}(x^3 + 5x + 1)$$

Applying the power rule, we get:
$$f'(x) = 3x^2 + 5$$

Now, look at this expression. We know that for any real number $x$, the square $x^2$ is always non-negative, meaning $x^2 \ge 0$.

Consequently, $3x^2 \ge 0$, and adding $5$ ensures that $f'(x) \ge 5$.

Since the derivative is always strictly positive, the slope of the function is always positive. The function is strictly increasing and never turns back; therefore, it is **one-one**.

### The Onto Journey: Exploring the Limits

Now, let us tackle the 'no-gap' rule. We need to see if the range of $f(x)$ covers all real numbers $\mathbb{R}$.

Since $f(x)$ is a polynomial, it is continuous everywhere. Let us explore the boundaries:
$$ \lim_{x 	o \infty} (x^3 + 5x + 1) = \infty $$
$$ \lim_{x 	o -\infty} (x^3 + 5x + 1) = -\infty $$

Because the function is continuous and travels from $-\infty$ to $\infty$, it must pass through every single real number in between. There are no gaps.

The range is $(-\infty, \infty)$, which is exactly the codomain $\mathbb{R}$. Thus, the function is **onto**.

### The Final Synthesis

We have walked through the logic. We proved the function is strictly increasing, confirming it is one-one.

We analyzed the limits at infinity, confirming it is onto. A function that is both one-one and onto is a **bijection**.

You have just mastered the behavior of a cubic polynomial. Keep this clarity with you; it is the key to solving even more complex problems in your JEE journey.

Animated Solution for Mathematics - Functions: For real $x$ , let $f (x) = x^{3} + 5 x + 1$ , then

Select Answer:

Visualized Solution

$f (x) = x^{3} + 5 x + 1$

Test for One-One

Setting up $f^{'} (x)$

Differentiating the Terms

Analyzing the Sign of $x^{2}$

Analyzing the Sign of $f^{'} (x)$

Conclusion on Injectivity

Test for Onto

Behavior at Positive Infinity

Behavior at Negative Infinity

Conclusion on Surjectivity

Final Answer

The Sigma Insight: Classification of Functions

Analyzing the Setup

The One-One Mystery

The Onto Journey

The Final Synthesis

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Animated Solution for Mathematics - Functions: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For real x, let f(x)=x3+5x+1, then