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Home » Open Your Eyes: The Untold Truth About the Most Controversial Ugly Face! - Carbonext

Open Your Eyes: The Untold Truth About the Most Controversial Ugly Face! - Carbonext

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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📌 \text{proj}_{\mathbf{a}} \mathbf{r} = \frac{\mathbf{r} \cdot \mathbf{a}}{\mathbf{a} \cdot \mathbf{a}} \mathbf{a}

📌 \mathbf{r} \cdot \mathbf{a} = 2 \cdot 1 + 3 \cdot 1 + 6 \cdot 1 = 11

📌 \mathbf{a} \cdot \mathbf{a} = 1^2 + 1^2 + 1^2 = 3

📌 Thus, the projection is:

📌 \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 \(x\):

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