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45 70 government rounds - ECD Germany

📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 Let \( u = x^2 + 2 \). Then, \( h(u) = 3\sqrt{u - 2} + 5 \). We want to find \( h(x^2 - 2) \), so set \( u = x^2 - 2 \). Then,

📌 \[ h(x^2 - 2) = 3\sqrt{(x^2 - 2) - 2} + 5 = 3\sqrt{x^2 - 4} + 5. \]

📌 Thus, \( h(x^2 - 2) \) is:

📌 \boxed{3\sqrt{x^2 - 4} + 5}

📌 Question: For all real numbers \( a, b, c \), find the number of functions \( f: \mathbb{R} \to \mathbb{R} \) such that \( f(a + b) = f(a) + f(b) \) and \( f \) is linear.

📌 Solution: A function \( f: \mathbb{R} \to \mathbb{R} \) is linear if it can be expressed as \( f(x) = kx \) for some constant \( k \). Given the functional equation:

📌 \[ f(a + b) = f(a) + f(b), \]

📌 assuming linearity, the function satisfies this by the very definition of linearity (additivity and homogeneity). Thus, the only functions meeting the criteria are those of the form \( f(x) = kx \). Since \( k \) can be any real number, there are infinitely many such linear functions, one for each real number \( k \). Therefore, the number of such functions is:

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