A Sine-Like Functional Equation

★ 7/10functions Recursion Functional Equations Symmetry

Problem

Functions $f(x)$ and $g(x)$ are defined on $\mathbb{R}$ and satisfy

f(x - y) = f(x)g(y) - g(x)f(y), \qquad f(-2) = f(1) \neq 0, \qquad g(-1) = g(2).

Determine, with proof, which of the following are true:

1. $g(0) = 1$ ; 2. the graph of $f(2x - 1)$ is symmetric about the point $\left(\dfrac{1}{2}, 0\right)$ ; 3. $g(1) + g(-1) = 1$ ; 4. if $f(1) = \dfrac{\sqrt{3}}{2}$ , then $\displaystyle\sum_{k=1}^{2023} f(k) = \dfrac{\sqrt{3}}{2}$ .

Answer

Solution

Difficulty	7/10
Topics	functions, Recursion, Functional Equations, Symmetry

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