Let f: R →R, f(x-f(y))=f(f(y))+x f(y)+f(x)-1 ∀x, y ∈R, if f(0)=1 and f^'(0)=0, then (A) f(x)=1-(

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Let f: R →R, f(x-f(y))=f(f(y))+x f(y)+f(x)-1 ∀x, y ∈R, if...

Let $f: R \rightarrow R, f(x-f(y))=f(f(y))+x f(y)+f(x)-1 \forall x$,
$y \in R$, if $f(0)=1$ and $f^{\prime}(0)=0$, then
(A) $\mathrm{f}(\mathrm{x})=1-\frac{\mathrm{x}^{2}}{2}$
(B) $f(x)=x^{2}+1$
(C) $f(x)=\left(\frac{2 x+1}{x+1}\right)$
(D) none of these

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Key Concepts

Functional Equations

Functional equations involve finding functions that satisfy an equation involving the function itself evaluated at different points. The process often involves identifying invariant properties and testing the function with strategically chosen inputs.

Substitution Method

The substitution method is a technique where specific values are substituted into the equation to simplify and extract important information about the function. By substituting values like 0 or expressions that cancel terms, one can deduce constraints on the form of the function.

Initial Conditions

Initial conditions, such as given function values and derivative values at a particular point, play a critical role in determining a unique solution among many possible functions that satisfy the functional equation. They serve to eliminate extraneous solutions and guide the function towards its correct form.

Quadratic Functions

Quadratic functions, which are second-degree polynomials, often appear as solutions to functional equations when the structure of the problem suggests a polynomial behavior. Their well-known properties, including symmetry and specific derivative characteristics, make them useful candidates in problems where the function and its derivative are specified.

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