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How many solutions does the equation have?\(Let\[f(x) = \left\{ \begin{array}{cl} -x + 3 & \text{if } x \le 0, \\ 2x - 5 & \text{if } x > 0. \end{array} \right.\]\)

 Oct 28, 2023
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Ifx≤0, then [f(f(x)) = f(-x + 3) = 2(-x + 3) - 5 = -2x + 1.]If x>0, then [f(f(x)) = f(2x - 5) = -2x - 5 + 3 = -2x - 2.]Hence,our problem is reduced to the two equations \begin{align*} -2x + 1 &= 4\ -2x - 2 &= 4. \end{align*}The first equation is solved byx=23​,and the second equation is solved byx=−3.Since each solution satisfies the corresponding inequality,there are2​solutions.

 Oct 28, 2023

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