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Q.
The domain of the function $f \left(x\right) = exp \left(\sqrt{5x-3-2x^{2}}\right)$ is
Relations and Functions
Solution:
We have, $f (x) =$ exp $\left(\sqrt{5x-3-2x^{2}}\right)$
i.e., $f \left(x\right) = e^{\sqrt{5x-3-2x^2}}$
For Domain of $f \left(x\right), \sqrt{5x-3-2x^{2}}$ should be +ve.
i.e., $\sqrt{5x-3-2x^{2}} \ge 0\quad$ (By taking -ve sign common)
$\Rightarrow \quad2x\left(x -1\right) -3\left(x -1\right) \le 0$
$\Rightarrow \quad \left(2x-3\right)\left(x -1\right) \le 0$
$\Rightarrow \quad2x - 3 \le 0\quad$ or $\quad x -1\ge 0$
$\Rightarrow \quad x \le \frac{3}{2}\quad$ or $\quad x \ge 1$
$\therefore 1 \le x \le \frac{3}{2}\quad$ i.e., $\quad x \in \left[1, \frac{3}{2}\right]$
Hence, domain of the given function is $[1, \frac{3}{2}].$