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\( a=\frac{4x^{2}-1}{2x^{2}-3|x|+1} \)

Ask by Alexander Schneider. in Greece
Jan 20,2025

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Η συνάρτηση \( a = \frac{4x^{2} - 1}{2x^{2} - 3|x| + 1} \) μπορεί να εκφραστεί παραγοντοποιημένα ως: \[ a = \begin{cases} \frac{2x + 1}{x - 1} & \text{για } x \geq 0, \\ \frac{2x - 1}{x + 1} & \text{για } x < 0. \end{cases} \] Σημειώσεις: - Οι σημεία \( x = \frac{1}{2} \) και \( x = -\frac{1}{2} \) πρέπει αποφεύγειν για να διατηρείται η συνάρτηση ορισμένη. - Η συνάρτηση παρουσιάζει ασυνέχειες στα αναφερθέντα σημεία.

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When we dive into the expression \( a=\frac{4x^{2}-1}{2x^{2}-3|x|+1} \), it's interesting to note that the numerator is a difference of squares! You can factor \( 4x^2 - 1 \) as \( (2x-1)(2x+1) \). Now, let’s talk about the denominator. The expression \( 2x^{2}-3|x|+1 \) varies depending on whether \( x \) is positive or negative, as the absolute value affects the linear term. For positive \( x \), it simplifies to \( 2x^2 - 3x + 1 \), whereas for negative \( x \), it becomes \( 2x^2 + 3x + 1 \). This gives you an opportunity to handle each case separately!

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