# What is the domain and range of x = y^2 -9?

Nov 8, 2017

"D:" {x inRR | x >= ~9}.
$\text{R:} \left\{y \in \mathbb{R} | y \ge 0\right\}$.

#### Explanation:

Rather than just say the domain and range, I'll show you how I got the answer, step by step.

First off, let's isolate $y$.

$x = {y}^{2} - 9$

$x + 9 = {y}^{2}$

$\sqrt{x + 9} = y$

Now, we can identify the type of function.

Let's describe the transformations of the function before we go on to the domain and range.

$y = \sqrt{x + 9}$

• There is only a horizontal translation of $9$ units to the left.

Now that that is done with, let's graph the function, so it is easier to determine the domain and range. Graphing is not necessary, but it makes it much easier.

The easiest way to graph this function is to sub in values for $x$ and solve for $y$. Graph the variables you subbed and solved for.

graph{y=sqrt(x+9) [-10, 10, -5, 5]}

We can see that the domain can only be values that are equal to or greater than ~9, thus, the domain is {x inRR | x >= ~9}.

As for the range, it can only be values that are equal to or greater than $0$, thus, the range is $\left\{y \in \mathbb{R} | y \ge 0\right\}$.

Hope this helps :)