Problem 163 - A Complete Resource Book in Mathematics for JEE Main

Question

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

Option (C): \(f\) is one-one and onto \(\mathbb{R}\).

Step 1: Understanding the Problem

We need to analyze the function \( f(x) = x^3 + 5x + 1 \) to determine if it is injective (one-one) and surjective (onto) over the set of real numbers \( \mathbb{R} \).

Step 2: Testing if the Function is One-One

A function is one-one (injective) if different inputs produce different outputs. To test this, we set \( f(a) = f(b) \) for arbitrary real numbers \( a \) and \( b \), resulting in the equation: \[ a^3 + 5a + 1 = b^3 + 5b + 1. \] After simplifying, we get \( a^3 - b^3 + 5(a - b) = 0 \). Factorize the expression to get \((a - b)(a^2 + ab + b^2 + 5) = 0\). For the product to be zero, either \( a - b = 0 \) or \( a^2 + ab + b^2 + 5 = 0 \). The expression \( a^2 + ab + b^2 + 5 \) is always positive for real \( a \) and \( b \), so \( a - b = 0 \), indicating \( a = b \). Hence, the function is injective.

Step 3: Testing if the Function is Onto

A function is onto (surjective) if every real number \( y \) can be expressed as \( f(x) \) for some real \( x \). In this scenario, we need to find if for every \( y \in \mathbb{R} \), there exists an \( x \) such that \( y = x^3 + 5x + 1 \). However, considering the limits of \( f(x) \) as \( x \) approaches \( \pm \infty \), the function approaches \( \pm \infty \) accordingly. This behavior and its derivative (which doesn't change sign) indicate that the range is the entire set of real numbers, confirming that the function is surjective.

Step 4: Conclusion from Analysis

From the above analysis, we conclude that \( f(x) = x^3 + 5x + 1 \) is both one-one and onto for real \( x \), which matches option (C): \( f \) is one-one and onto \( \mathbb{R} \).

Key Concepts

Cubic functions

Cubic functions are a fascinating type of polynomial function, often expressed in the general form \( f(x) = ax^3 + bx^2 + cx + d \). In the exercise, the function \( f(x) = x^3 + 5x + 1 \) is a cubic function with no \( x^2 \) term. This means that its graph will be a curve with specific characteristics:

• It has one turning point (a local maximum or minimum) and may change direction only once.
• The leading coefficient (in this case 1) is positive, which implies that the curve starts from the lower left and rises to the upper right as \( x \) increases.
• Because it is a polynomial of odd degree (3 here), a cubic function will always have at least one real root.

The task was to understand how this particular function behaves when evaluated at different values of \( x \), and to determine if it is injective (one-one) and surjective (onto). The real test was to explore the function's inherent properties like its general shape and roots to conclude what every \( x \) and \( f(x) \) pair implies.

Real analysis

Real analysis is the study of real numbers and real-valued functions, providing a theoretical backbone for examining functions like the one seen in the exercise. To determine whether the function \( f(x) = x^3 + 5x + 1 \) is injective and surjective, several real analysis concepts come into play.

• Injective Function (One-One): This type of function means every element of the domain maps to a unique element in the codomain. We used an algebraic approach to test injectivity by setting \( f(a) = f(b) \) and seeing if this implies \( a = b \). A factorization confirmed that this cubic function is indeed injective, as the derived expression only holds for equal inputs \( a \) and \( b \).
• Surjective Function (Onto): A function is surjective if, for every real number \( y \), there is some real number \( x \) such that \( f(x) = y \). In our scenario, because the function can reach every real value as \( x \) approaches either positive or negative infinity, it's established that the function is also surjective. This part required understanding the behavior of the function's limits, which is rooted in real analysis' exploration of continuity and infinity.

Thus, real analysis helps verify formal proofs and lend a deeper understanding of functions by scrutinizing properties like limits, roots, and behaviors.

Function transformations

Function transformations describe how the graph of a basic function can be shifted, stretched, or otherwise altered to produce a new, transformed graph. This concept is significant in understanding cubic functions like \( f(x) = x^3 + 5x + 1 \). Let's explore some key transformations:

• Translation & Shifting: Adding a constant term to a function, as seen with the \( +1 \) in our cubic function, vertically shifts the entire graph upward by one unit.
• Scalings and Stretching: The coefficient of \( x^3 \) is 1, meaning no vertical stretching or shrinking occurs, and the shape of the original \( x^3 \) is preserved. However, the \( 5x \) term affects the slope of the curve, slightly modifying the steepness and turning points.
• Reflection: Although not present in this specific function, reflections occur if negative signs are introduced. For example, if \( f(x) = -x^3 + 5x + 1 \), the graph would be reflected across the x-axis, producing an entirely different orientation.

By observing these transformation effects, one can predict the graph's behavior and compare it to a base cubic function, thereby understanding how specific terms impact the overall function.