# How do you determine the domain and range of a function?

May 25, 2018

See below

#### Explanation:

I will assume $\left\{f \left(x\right) , x\right\} \in \mathbb{R}$

Then, the domain of $f \left(x\right)$ is the set of all real values of $x$ for which $f \left(x\right)$ is defined. We can think of this as the valid inputs. Let's now call this set $D$

Then the range of $f \left(x\right)$ is the set of values of $f \left(x\right)$ over $D$. We can think of this as the valid outputs.

To determine the domain and range of a function, first determine the set of values for which the function is defined and then determine the set of values which result from these.

E.g. $f \left(x\right) = \sqrt{x}$

$f \left(x\right)$ is defined $\forall x \ge 0 : f \left(x\right) \in \mathbb{R}$

Hence, the domain of $f \left(x\right)$ is $\left[0 , + \infty\right)$

Also, $f \left(0\right) = 0$ and $f \left(x\right)$ has no finite upper bound.

Hence, the range of $f \left(x\right)$ is also $\left[0 , + \infty\right)$

We can deduce these results from the graph of $\sqrt{x}$ below.

graph{sqrtx [-4.18, 21.13, -6.51, 6.15]}