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Definition of Functions

Find the doma...

Question

Find the domain of the following function.
$y = \sqrt{- 4 x^{2} + 4 x + 3} .$

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Solution

$y = \sqrt{- {4 x}^{2} + 4 x + 3}$
$(- {4 x}^{2} + 4 x + 3) \geq 0$
$(- {4 x}^{2} + 6 x - 2 x + 3) \geq 0$
$2 x (- 2 x + 3) + 1 (- 2 x + 3) \geq 0$
$(2 x + 1) (- 2 x + 3) \geq 0$
$x \geq - \frac{1}{2}$
$x \leq \frac{3}{2}$
So, the domain of $y = [- \frac{1}{2}, \frac{3}{2}]$

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