Question: A brain-computer interface algorithm uses a function $ f $ satisfying $ f(x + y) + f(x - y) = 2f(x) + 2f(y) $ for all real $ x, y $. If $ f(1) = 3 $, compute $ f(5) $.
["Understanding Brain-Computer Interfaces Through Advanced Mathematical Models: Solving the Functional Equation for $ f $", "In the rapidly evolving field of brain-computer interfaces (BCIs), mathematical models play a crucial role in decoding signals between the human brain and external devices. Interesting algorithmic foundations often rely on elegant functional equations derived from neurophysiological data. One such equation is:", "$$\nf(x + y) + f(x - y) = 2f(x) + 2f(y) \quad \ ext{for all } x, y \in \mathbb{R}\n$$", "This functional equation, widely studied in harmonic analysis and applied mathematics, describes symmetric signal processing behaviors relevant to BCI algorithms. Given $ f(1) = 3 $, we aim to compute $ f(5) $ using properties of $ f $, avoiding brute-force iteration by leveraging known structural insights.", "---", "### Step 1: Recognize the Functional Equation", "The given equation:", "$$\nf(x + y) + f(x - y) = 2f(x) + 2f(y)\n$$", "is a classical functional equation. Functions satisfying it over $ \mathbb{R} $ are known to be quadratic polynomials under mild regularity conditions (e.g., continuity, measured monotonicity). That is, $ f(x) = ax^2 $ is a general solution.", "Let us verify this: suppose $ f(x) = ax^2 $. Then:", "$$\nf(x + y) = a(x + y)^2 = a(x^2 + 2xy + y^2) \\nf(x - y) = a(x - y)^2 = a(x^2 - 2xy + y^2)\n$$", "Adding:", "$$\nf(x + y) + f(x - y) = a(2x^2 + 2y^2) = 2ax^2 + 2ay^2 = 2f(x) + 2f(y)\n$$", "So the equation holds. Therefore, all solutions are of the form $ f(x) = ax^2 $.", "---", "### Step 2: Use Initial Condition to Determine $ a $", "We are given $ f(1) = 3 $. Substituting into $ f(x) = ax^2 $:", "$$\nf(1) = a(1)^2 = a = 3 \quad \Rightarrow \quad a = 3\n$$", "Thus, the function is:", "$$\nf(x) = 3x^2\n$$", "---", "### Step 3: Compute $ f(5) $", "Now substitute $ x = 5 $:", "$$\nf(5) = 3 \cdot 5^2 = 3 \cdot 25 = 75\n$$", "---", "### Final Insight: From Math to Brain-Computer Interfaces", "In BCI applications, such functional relationships model spatial or temporal signal propagation in neural networks. The quadratic form reflects energy-like transfer properties in neural feedback loops, where squared dependence on input amplitude ensures stable, predictable signal amplification. Knowing $ f(1) = 3 $ allows fast, deterministic scaling—critical for real-time decoding.", "---", "Conclusion:\nUsing the structure of the functional equation and the given value $ f(1) = 3 $, we found $ f(x) = 3x^2 $. Therefore,\n$$\nf(5) = 75\n$$", "This elegant solution underscores how mathematical insights power cutting-edge neurotechnology.", "---", "Keywords: brain-computer interface, functional equation, $ f(x + y) + f(x - y) = 2f(x) + 2f(y) $, $ f(1) = 3 $, $ f(5) $, quadratic function, quadratic interpolation, neural signal modeling."]
