Problem 5 - Calculus 1 Course in Mathematics for the IIT JEE and Other Engineering Exams

Question

If for a function \(f(x), f(x+y)=f(x)+f(y)\) for all reals \(x\) and \(y\) then \(f(0)\) is equal to (a) 1 (b) 0 (c) \(f(x) \forall x \in R\) (d) None of these

Step-by-Step Solution

Answer

(b) 0

Step 1: Understand the Functional Equation

We are given that for a function \(f(x)\), \(f(x+y)=f(x)+f(y)\) for all real numbers \(x\) and \(y\). This is a well-known property of linear functions and is called a Cauchy functional equation.

Step 2: Substitute Specific Values

To find \(f(0)\), start by setting \(x = 0\) and \(y = 0\) in the functional equation: \(f(0 + 0) = f(0) + f(0)\). This simplifies to \(f(0) = 2f(0)\).

Step 3: Solve for \(f(0)\)

The equation \(f(0) = 2f(0)\) implies that \(2f(0) - f(0) = 0\), or \(f(0) = 0\).

Step 4: Conclusion

Since no other value satisfies the equation \(f(0) = 2f(0)\) other than \(f(0) = 0\), \(f(0)\) must be 0.

Key Concepts

Functional Equations

Functional equations describe relationships between functions, often involving the values of the functions at various points. One famous example is the Cauchy functional equation. In this problem, it is expressed as \(f(x + y) = f(x) + f(y)\), which implies a specific structure of the function involved. These types of equations are great for uncovering information about a function when specific algebraic rules apply.

When dealing with functional equations:

• Consider substituting specific values to simplify the equation.
• Observe how the equation behaves under these specific conditions.
• Look for invariants or properties that the function must satisfy.

Substituting values helps break down the equation into simpler parts, often revealing hidden constants or symmetries. In the given Cauchy equation, setting specific values helps quickly identify that \(f(0) = 0\), revealing an essential property of the function under this equation. Solving functional equations often involves recognizing patterns and properties like these.

Linear Functions

Linear functions are a fundamental concept in mathematics, defining functions of the form \(f(x) = mx + b\), where \(m\) and \(b\) are constants. The Cauchy functional equation, \(f(x+y) = f(x) + f(y)\), is closely related to linear functions, particularly those that are homogenous linear functions with \(b = 0\).

Linear functions are characterized by:

• A constant rate of change, represented by \(m\), the slope of the line.
• A graph that is a straight line.
• The property that for any inputs \(x\) and \(y\), the function’s value changes linearly.

For a function satisfying the Cauchy equation, it suggests that \(f(x)\) could indeed be linear but without a constant \(b\) term, such as \(f(x) = cx\) where \(c\) is a constant, which simplifies further in specific cases to \(f(x) = 0\) for all \(x\) when \(f(0) = 0\). Understanding the influence of linearity helps in comprehending solutions to functional equations that exhibit patterns similar to the Cauchy form.

Real Numbers

Real numbers (\(R\)) form the basis for many mathematical equations and functions, including functional equations like the Cauchy equation. They are a complete ordered field, meaning they include all the positive and negative rational numbers, zero, and the irrational numbers, creating a continuous numerical line with no gaps.

Key aspects of real numbers:

• The set of real numbers is denoted as \(R\).
• Real numbers can be visualized on the number line, extending infinitely in both directions.
• They include distinct subsets such as rational and irrational numbers.

In functional equations, real numbers allow functions such as \(f(x +y) = f(x) + f(y)\) to be evaluated at any point on this continuous spectrum. This continuity is crucial for analyzing and solving equations since evaluating at any number allows for flexibility in testing and verifying the properties of functions. Solving the Cauchy functional equation effectively emphasizes this by testing the universality of functional properties across all real numbers.