For what values of $x$ will the expression represent a real number? Write answers using inequality notation.
$\sqrt{3 x^{2} - 7 x - 6}$

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Solution

$D o m a i n o f \sqrt{3 x^{2} - 7 x - 6} : x \leq - \frac{2}{3} o r x \geq 3$

$R a n g e o f \sqrt{3 x^{2} - 7 x - 6} : [\begin{matrix} S o l u t i o n : & f (x) \geq 0 I n t e r v a l N o t a t i o n : & [0, \infty) \end{matrix}]$

$E x t r e m e P o i n t s o f \sqrt{3 x^{2} - 7 x - 6} : M i n i m u m (- \frac{2}{3}, 0), M i n i m u m (3, 0)$

$f^{'} (x) = \frac{6 x - 7}{2 \sqrt{3 x^{2} - 7 x - 6}}$

$T h e c r i t i c a l p o i n t s : x = - \frac{2}{3}, x = 3$

$T h e r a n g e f o r t h e i n t e r v a l - \infty < x \leq - \frac{2}{3} : 0 \leq f (x) < \infty$

$T h e r a n g e f o r t h e i n t e r v a l 3 \leq x < \infty : 0 \leq f (x) < \infty$

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