MathDB

We say that a function

f: \mathbb{R} \to \mathbb{R}

is morally odd if its graph is symmetric with respect to a point, that is, there exists

(x_0, y_0) \in \mathbb{R}^2

such that if

(u, v) \in \{(x, f(x)) : x \in \mathbb{R}\}

, then

(2x_0 - u, 2y_0 - v) \in \{(x, f(x)) : x \in \mathbb{R}\}

. On the other hand,

f

is said to be morally even if its graph

\{(x, f(x)) : x \in \mathbb{R}\}

is symmetric with respect to some line (not necessarily vertical or horizontal). If

f

is morally even and morally odd, we say that

f

is parimpar.
(a) Let

S \subset \mathbb{R}

be a bounded set and

f: S \to \mathbb{R}

be an arbitrary function. Prove that there exists

g: \mathbb{R} \to \mathbb{R}

that is parimpar such that

g(x) = f(x)

for all

x \in S

.
(b) Find all polynomials

P

with real coefficients such that the corresponding polynomial function

P: \mathbb{R} \to \mathbb{R}

is parimpar.

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