# Domains

The domain of a function is the set of all values for which the function is defined.

For most functions in algebra, the domain is the set of all real numbers .  But, there are two cases where this is not always true, fractions with a variable in the denominator and radicals with an even index.

Example 1:

Find the domain of $f\left(x\right)=\frac{x+3}{x-2}$ .

Since division by zero is undefined in the real number system, $x\ne 2$ .  So the domain is all real numbers except $2$

Example 2:

Find the domain of $f\left(x\right)=\sqrt{x-2}$ .

Since we can only take the square root of a non-negative number, the domain is all real numbers greater than or equal to $2$

$x\ge 2$

You may sometimes be presented with an equation and a domain of possible solutions. In this case the domain means the set of possible values for the variable.

Example 3:

Solve the equation

${x}^{2}=\sqrt{x}$

over the domain $\left\{0,1,2,3\right\}$ .

This is a tricky equation; it's not linear and it's not quadratic , so we don't have a good method to solve it. However, since the domain only contains four numbers, we can just use trial and error.

$\begin{array}{l}{0}^{2}=\sqrt{0}=0\\ {1}^{2}=\sqrt{1}=1\\ {2}^{2}\ne \sqrt{2}\\ {3}^{2}\ne \sqrt{3}\end{array}$

So the solution set over the given domain is $\left\{0,1\right\}$ .