Find the domain of the function $f(x) = \sqrt{6-x-x^2-2x^2}$.

Answer 1

For $f(x)$ to be defined, we need $6-x-x^2-2x^2$ to be nonnegative. We can write this as \[9 - 3x - 3x^2 \ge 0.\]The expression on the left factors as $-(3x+3)(x-1)$. Plotting the solutions $x=-1$ and $x=1$, we find the function is nonnegative for $x \le -1$ and $1 \le x$, so the domain of $f(x)$ is $\boxed{(-\infty,-1] \cup [1,\infty)}$.

[asy]
unitsize(1.5 cm);

draw((-2.5,0)--(2.5,0));
draw((0,-1.5)--(0,1.5));
draw((-1.5,-0.5)--(-1.2,0));
draw((0.5,0)--(2,0.8));
draw(circle((-1,0.2),0.06));
draw(circle((1,0),0.06));

label("$1$", (1,0), N);
label("$-1$", (-1,0), N);
label("$6 - x - x^2 - 2x^3 = 0$", (2,0.8), E);
[/asy]