Which of the following functions satisfies f(x+y)=f(x)+f(y) for all real numbers x and y ? (a) f

step 1 So for this problem, we are trying to test to see which function addition gives us the following

Which of the following functions satisfies f(x+y)=f(x)+f(y)...

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Key Concepts

Cauchy's Functional Equation

Cauchy's functional equation is the equation f(x + y) = f(x) + f(y) which defines additive functions. Under additional conditions, such as continuity or boundedness on an interval, the only solutions to this equation are functions of the form f(x) = ax, where a is a constant. This concept is central to identifying functions that satisfy the equation.

Linear Functions

Linear functions are functions that can be expressed in the form f(x) = ax, where a is a constant. Such functions naturally satisfy the additivity property because the sum f(x) + f(y) equals a(x + y), which is exactly f(x + y). Recognizing the structure of linear functions helps in distinguishing valid solutions from non-linear or affine functions.

Role of Constant Terms

When a function includes a constant term, for example in an affine function of the form f(x) = ax + b where b is nonzero, the additivity property generally fails. The constant term does not cancel out when computing f(x + y), which is why only functions without such a term (i.e., pure linear functions) satisfy the equation.

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