Question

What is the Domain of the function $f(x)=\sqrt{2-2x-2x^{2} }$

Answer 1

We're asked to find the domain of the function,

$\longrightarrow f(x)=\sqrt{2-2x-2x^2}$

Here the restriction arises that the polynomial inside the square root must be non - negative, i.e.,

$\longrightarrow 2-2x-2x^2\geq0$

Dividing by 2,

$\longrightarrow 1-x-x^2\geq0$

Multiplying by -1, (note the symbol change)

$\longrightarrow x^2+x-1\leq0\quad\quad\dots(1)$

Let us find the roots of the equality, $x^2+x-1=0.$

$\longrightarrow x=\dfrac{-1\pm\sqrt{1^2-4\times1\times-1}}{2\times1}$

$\longrightarrow x=\dfrac{-1\pm\sqrt5}{2}$

Hence the solution to the inequality (1) is,

$\longrightarrow\underline{\underline{x\in\left[\dfrac{-1-\sqrt5}{2},\ \dfrac{-1+\sqrt5}{2}\right]}}$

or,

$\longrightarrow\underline{\underline{x\in\left[-\dfrac{\sqrt5+1}{2},\ \dfrac{\sqrt5-1}{2}\right]}}$

This is the domain of our function.