Problem 70 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

For real \(x\), let \(f(x)=x^{3}+5 x+1\), 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 \(R\)

Step-by-Step Solution

Answer

The function \( f(x) = x^3 + 5x + 1 \) is one-one and onto \( \mathbb{R} \). (C)

Step 1: Understand the Function

The function given is \( f(x) = x^3 + 5x + 1 \). It is a polynomial function of degree 3, which means it is a cubic function.

Step 2: Investigate One-One Property

A function is one-one if each element of the domain maps to a unique element in the range. For a function to be one-one, its derivative must not change sign (i.e., it must be either always positive or always negative). Calculate \( f'(x) \): \[ f'(x) = 3x^2 + 5 \]. Since \( 3x^2 + 5 \) is always positive for all real \( x \) (because \( 3x^2 \) is non-negative and \( 5 \) is positive), \( f(x) \) is strictly increasing. Thus, \( f(x) \) is one-one.

Step 3: Investigate Onto Property

A function is onto if every possible element in the codomain is an image of some element in the domain. Since \( f(x) \) is a cubic polynomial, as \( x \to +\infty, f(x) \to +\infty \) and as \( x \to -\infty, f(x) \to -\infty \). Thus, \( f(x) \) can take all real numbers because it's continuous and strictly increasing with no breaks. Therefore, the function is onto \( \mathbb{R} \).

Step 4: Conclusion

From the analysis in Steps 2 and 3, we find that \( f(x) \) is both one-one (injective) and onto (surjective). Therefore, option (C) is the correct choice.

Key Concepts

Injective Function

An injective function, also known as a one-to-one function, is characterized by mapping each element of its domain to a unique element in the range. For students, this means there are no two different inputs that result in the same output. If you imagine the function as a machine, no two different items can produce the exact same result once they've been processed by the machine.

To determine if a function, like our cubic function \(f(x) = x^3 + 5x + 1\), is injective, we use the derivative of the function. The derivative, \(f'(x) = 3x^2 + 5\), helps us understand if the function is consistently increasing or decreasing. In this case, since \(3x^2\) is non-negative and \(5\) is positive, \(f'(x)\) is always positive for all real \(x\).

This means no matter what value of \(x\) you choose, the function will always increase, showing that each \(x\) leads to a different \(f(x)\). Therefore, \(f\) is injective.

Surjective Function

A surjective function, or onto function, is one where every element in the codomain (the set of all possible outputs) corresponds to at least one element from the domain. In simpler terms, there aren't any outputs left out — every possible output from the function is covered by some member of the domain.

For the cubic function \(f(x) = x^3 + 5x + 1\), we need to show that it is surjective, mapping onto all real numbers. Cubic functions have the property that as \(x\) approaches positive infinity, \(f(x)\) also approaches positive infinity. Similarly, as \(x\) approaches negative infinity, \(f(x)\) approaches negative infinity.

Since \(f(x)\) is both continuous and strictly increasing, there are no gaps in the outputs. This means that \(f\), being a cubic polynomial, can produce every real number, proving it to be a surjective function.

Polynomial Derivatives

Polynomial derivatives are crucial in analyzing the behavior of polynomial functions. Derivatives, in essence, provide the rate at which the function’s value changes with respect to changes in its input. For a polynomial function, the derivative is another polynomial.When considering the cubic function \(f(x) = x^3 + 5x + 1\), its derivative is calculated as \(f'(x) = 3x^2 + 5\). Here's what this derivative tells us:

• The term \(3x^2\) indicates a parabolic shape that opens upwards, and since \(x^2\) is always non-negative, the "+5" ensures the entire derivative is greater than zero for all real \(x\).
• Since \(f'(x)\) is always positive, \(f(x)\) is strictly increasing. This strict increase helps establish that the function is one-to-one, as no output is repeated.

This property of derivatives can be a powerful tool for students determining injectivity and monotonicity. Understanding how a derivative behaves provides critical insights into the function's global structure and behavior.