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Ask by Burton Romero. in Iran

Feb 04,2025

To find the range of \( a = |x^{2}-3|-2 \) for \( -2 < x < 1 \), first, let’s determine the values of \( x^2 \) in the given range. Since \( -2 < x < 1 \), we have \( 0 < x^2 < 1 \) when \( -2 < x < 1 \). Now, substituting \( x^2 \) within the absolute value function: 1. When \( x^2 < 3 \) (which is always true for the given range), we have: \[ a = |x^2 - 3| - 2 = 3 - x^2 - 2 = 1 - x^2 \] Thus, \( a \) simplifies to \( 1 - x^2 \). Next, let's find the extreme values of \( a \) by evaluating it at the boundaries of \( x \). - When \( x = -2 \), \( a = 1 - (-2)^2 = 1 - 4 = -3 \). - When \( x = 1 \), \( a = 1 - (1)^2 = 1 - 1 = 0 \). Given that \( x \) can take values between \( -2 \) and \( 1 \), the minimum value of \( a \) is \( -3 \) and the maximum is \( 0 \). Therefore, the range of \( a \) is: \[ -3 < a < 0. \]