Mathematics

Problem

If a non-zero function

f(x)

is symmetrical about

y = 1

then the value of

p

(constant) such that

f^2(x) = (f^{-1}(x))^2 - px \cdot f(x) \cdot f^{-1}(x) + 2x^2 f(x)

for all

x \in \mathbb{R}

is .....

Intuitive explanation

When a function is symmetric about the horizontal line $y = 1$ , it means $f(2 - x) = 2 - f(x)$ —the graph mirrors across that line. This constraint lets you relate $f$ and its inverse in a special way. The key is to use this symmetry property to substitute into the given functional equation and find what value of $p$ makes it true for all $x$ , which typically means comparing coefficients or testing strategic values.

Intuitive explanation

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