Question

Please help? Graph 12y^2=84" align="absmiddle" class="latex-formula">. Find domain and range.
I have the graph, but I don't know how to find the domain and range?

Answer 1

$-3x^2+12y^2=84$
$12y^2=3x^2+84$
$y^2=\dfrac{x^2}4+7$
$y=\pm\sqrt{\dfrac{x^2}4+7}$

For either square root to exist, you require that $\dfrac{x^2}4+7\ge0$. This is true for all $x$, since $\dfrac{x^2}4$ is always non-negative. This means the domain of $y$ as a function of $x$ is all real numbers, or $x\in\mathbb R$ or $(-\infty,infty)$.

Now, because $\dfrac{x^2}4+7$ is non-negative, and the smallest value it can take on is 7, it follows that the minimum value for the positive square root must be $\sqrt7$, while the maximum value of the negative root must be $-\sqrt7$. This means the range is $y\in\mathbb R\setminus(-\sqrt7,\sqrt7)$, or $|y|\ge\sqrt7$, or $(-\infty,-\sqrt7]\cup[\sqrt7,\infty)$.