Domains

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 $2$ . 

$x\ge 2$

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 $\left\{0,1\right\}$ .