Question

What is the domain and range of $y = \sqrt {{x^2} - 9}$ ?

Answer 1

Here we need to find the domain and range of the given square root function.
The domain is nothing but the set of all possible inputs and the range is the set of all its outputs of the given function.
To find the domain and range of a function, we need to first determine the set of values for which the function is defined and then determine the set of values that result from these.

Complete answer:
One way to find it is by graphing it and then looking for the $y$ axis which shows the range, and the $x$ axis which shows the domain.
Let’s solve this algebraically,
We know that the square root function is only defined when the number or the given expression under the radical sign is greater than or equal to $0$ .
Therefore, $y = \sqrt {{x^2} - 9}$
$\Rightarrow {x^2} - 9 \geqslant 0$
Now, we set it equal to $0$ .
$\Rightarrow {x^2} - 9 = 0$
$\Rightarrow {x^2} = 9$
$\Rightarrow x = 3$ , $x = - 3$ .
Now we have these points. We need to call them boundary points of the real number line. So we have three intervals $\left( { - \infty , - 3} \right)$ , $\left( { - 3,3} \right)$ , and $\left( {3,\infty } \right)$ .
We need to choose a point in each interval and substitute it into the original equation ${x^2} - 9 \geqslant 0$ to check if it is valid or not, and each boundary point to see if it is in the domain or not. We find that $- 3$ and $3$ is defined $\left( { - \infty , - 3} \right)$ and $\left( {3,\infty } \right)$ is also defined but $\left( { - 3,3} \right)$ is not. Since $\left[ {} \right]$ means including, we get the domain is $\left( { - \infty - 3} \right] \cup \left[ {3,\infty } \right)$ and we put $\left( {} \right)$ around the infinity symbols because infinity is never reached, so it is not an included value.
For the range, just think about this the lowest value a square root function can give is $0$ , because the lowest point where the square root function is defined is when it is $0$ . $\sqrt 0$ is defined, whereas $\sqrt { - 0.000001}$ is not. So it is simple $\left[ {0,\infty } \right)$ and including $0$ .

Note: A function is a relationship between the x and y values, where each $x$ value or input has only one $y$ value or output.
Domain: all $x$ values or inputs that have an output of real $y$ values.
Range: the $y$ values or outputs of a function.