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Home » Solution: We begin by factoring the numerator: - Anadea

Solution: We begin by factoring the numerator: - Anadea

📅 May 14, 2026 👤 scraface

May 14, 2026

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📌 Question: Find the range of the function \(f(x) = rac{2\sin x + 3}{\sin x + 2}\) as \(x\) ranges over all real numbers such that \(\sin x

📌 e -2\) (which is always true since \(\sin x \in [-1,1]\)).

📌 Solution: Let \(y = rac{2\sin x + 3}{\sin x + 2}\). Solve for \(\sin x\):

📌 y(\sin x + 2) = 2\sin x + 3 \Rightarrow y\sin x + 2y = 2\sin x + 3

📌 We require \(\left|rac{3 - 2y}{y - 2}

📌 ight| \leq 1\)

📌 \left| rac{3 - 2y}{y - 2}

📌 ight| \leq 1 \Rightarrow (3 - 2y)^2 \leq (y - 2)^2

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