If a function f: R →R be such that f(x-f(y))=f(f(y)) +x f(y)+f(x)-1, ∀x, y ∈R, then f(x)= (A) (x

step 1 So in this question we have F of X minus F of Y is equal to F of F of Y plus X of F of Y plus F

If a function f: R →R be such that f(x-f(y))=f(f(y)) +x...

If a function $f: R \rightarrow R$ be such that $f(x-f(y))=f(f(y))$ $+x f(y)+f(x)-1, \forall x, y \in R$, then $f(x)=$
(A) $\frac{x^{2}}{2}-1$
(B) $\frac{x^{2}}{2}+1$
(C) $1-\frac{x^{2}}{2}$
(D) None of these

Key Concepts

Coefficient Comparison

Coefficient comparison is a method used when a function is assumed to have a certain algebraic form, such as a polynomial. By substituting the assumed form into the equation and equating the coefficients of similar powers of the variable on both sides, one can solve for the unknown constants. This systematic approach is particularly useful in verifying if a candidate function satisfies the functional equation.

Polynomial Functions

Polynomial functions, especially quadratics, are a common class of functions that often emerge as potential solutions to certain functional equations. When a functional equation suggests that the function might be expressed in a polynomial form, assuming a general polynomial (for instance, a quadratic) and comparing coefficients can aid in determining the specific parameters of the polynomial.

Substitution Method

The substitution method is a key technique used in solving functional equations. It involves choosing specific values for the variables to simplify the equation, eliminate complexities, and isolate the function. By setting particular values, one can often reduce the equation to a simpler form which helps in deducing the general behavior or form of the function.

Functional Equations

A functional equation is an equation in which the unknowns are functions rather than simple numerical values. Solving a functional equation typically involves finding all functions that satisfy a given relation for all inputs in a specified domain. This process often requires exploring the properties of the function and applying suitable substitutions to simplify or reveal hidden structures within the equation.

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