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Home » The 2 discoveries in the 19th century block can be arranged in \(2! = 2\) ways. - Londonproperty

The 2 discoveries in the 19th century block can be arranged in \(2! = 2\) ways. - Londonproperty

📅 March 14, 2026 👤 scraface

Mar 14, 2026

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📌 \text{proj}_{\mathbf{a}} \mathbf{r} = \frac{11}{3} \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} \frac{11}{3} \\ \frac{11}{3} \\ \frac{11}{3} \end{bmatrix}

📌 \boxed{\begin{bmatrix} \frac{11}{3} \\ \frac{11}{3} \\ \frac{11}{3} \end{bmatrix}}

📌 Question:** A palynologist is analyzing pollen distribution in a region and models the concentration as a function \(f(x) = \frac{2x^2 - 3x + 1}{x^2 + 1}\). Determine the range of \(f(x)\) as \(x\) varies over all real numbers.

📌 Solution:** To find the range of \(f(x) = \frac{2x^2 - 3x + 1}{x^2 + 1}\), set \(y = f(x)\) and solve for \(x\):

📌 y = \frac{2x^2 - 3x + 1}{x^2 + 1}

📌 Multiply both sides by \(x^2 + 1\):

📌 y(x^2 + 1) = 2x^2 - 3x + 1

📌 yx^2 + y = 2x^2 - 3x + 1 \quad \Rightarrow \quad (y - 2)x^2 + 3x + (y - 1) = 0

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