Question

6. Write the domain of the following real functions i) √1-2x-3x^2​

Answer 1

Step-by-step explanation:

We have, $\rm\:f(x)=\sqrt{1-2x-3{x}^{2}}$

Now, since, square root function is defined for only positives values, so,

we have,

$\rm1 - 2x - 3 {x}^{2} \geqslant 0$

$\rm \implies 3 {x}^{2} + 2x - 1 \leqslant 0$

$\rm \implies 3 {x}^{2} + 3x - x - 1 \leqslant 0$

$\rm \implies 3 x(x + 1)-1( x + 1) \leqslant 0$

$\rm \implies (3 x - 1)(x + 1) \leqslant 0$

$\rm \implies 3\bigg( x - \frac{1}{3} \bigg)(x + 1) \leqslant 0$

$\rm \implies \bigg( x - \frac{1}{3} \bigg)(x + 1) \leqslant 0$

$\rm \implies \: x \in \bigg [ - 1 , \frac{1}{3} \bigg ]$

$\rm\:\bold{Hence, \:\:domain\:\:\in \bigg [ - 1 , \frac{1}{3} \bigg ]}$