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Home » \( x^2 - 4x - 16 = 0 \) - Abu Waleed Tea

\( x^2 - 4x - 16 = 0 \) - Abu Waleed Tea

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 Let’s set $ a = b = c = 0 $:

📌 f(0) = 3f(0) \Rightarrow 2f(0) = 0 \Rightarrow f(0) = 0

📌 Now, set $ c = 0 $:

📌 This is the **Cauchy Functional Equation**, whose solutions over $ \mathbb{R} $ are linear functions $ f(x) = kx $, assuming some regularity condition (like continuity, monotonicity, or boundedness). Since the problem does not restrict $ f $, but asks for the **number** of such functions, we must consider all additive functions from $ \mathbb{R} \to \mathbb{R} $.

📌 Under the Axiom of Choice, there are **infinitely many** additive functions (including pathological ones), but if we restrict to **real-valued additive functions** (without further constraints), the number of such functions is uncountably infinite.

📌 However, in the context of a math olympiad, and given the phrasing find the number of functions, we assume the standard assumption of **linear solutions**, i.e., $ f(x) = kx $.

📌 Thus, the number of such functions is equal to the number of real constants $ k $, which is uncountably infinite. But if we are to count **distinct functions**, and assuming only linear solutions are acceptable, the number is infinite.

📌 However, if the problem is interpreted as asking for the number of **continuous** solutions, then the only such functions are linear: $ f(x) = kx $, and there are infinitely many such functions (one for each real $ k $).

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