# How do you find the domain and range of sqrt(x-3) ?

Dec 18, 2016

Domain$\to x \ge 3$

Range $\to y \ge 0$

#### Explanation:

$\textcolor{b l u e}{\text{Determine the domain}}$

There is a difference between the words can and may

You can square root a negative value but the answer belongs to the set of 'complex numbers'.

On the other hand, if you wish to avoid complex numbers the content of the root may not become negative.

Consequently I am setting the condition:$\text{ } x - 3 \ge 0$

$\textcolor{g r e e n}{\implies x \ge + 3} \textcolor{red}{\to \left\{x \in \mathbb{R} : x \ge 3\right\} \leftarrow \text{ bit of higher level maths}}$

$\left\{x \in \mathbb{R} : x \ge 3\right\}$ means: the set value of $x$ such that $x$ belongs to the set of 'Real Numbers' (not complex numbers) and that it is greater than or equal to 0.

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$\textcolor{b l u e}{\text{Determine the range}}$

Domain can be thought of as 'input'. So for this expression it is the values you assign to $x$

Range can be thought of as 'output' and it is all the positive values greater than or equal to 0

Set $y = \sqrt{x - 3}$

So the range is $y$ and

$\textcolor{g r e e n}{y \ge 0} \textcolor{red}{\to \left\{y \in \mathbb{R} : y \ge 0\right\} \leftarrow \text{bit of higher level maths}}$

or you could write: $\textcolor{red}{y \in \left[0 , \infty\right) \leftarrow \text{bit of higher level maths}}$

$\left[0 , \infty\right)$ means all the number from 0 to tending towards infinity. You can never really get to infinity so it is excluded. So the { is inclusive and the ) is exclusive.