Skill: Determine the domain of a function from an equation

The domain of a function is the set of all real numbers that can be used as inputs without making the equation undefined.

Check for these restrictions:

• A denominator cannot equal $0$.
• An even-root radicand must be greater than or equal to $0$.
• A logarithm’s argument must be greater than $0$.
• A polynomial has domain all real numbers.

Example

Determine the domain of

$$f(x)=\frac{\sqrt{x+2}}{x-3}.$$

Step 1: Check the square root

The expression inside the square root must be nonnegative:

$$x+2\ge 0$$

Subtract $2$:

$$x\ge -2.$$

So the square root allows inputs in the interval

$$[-2,\infty).$$

Step 2: Check the denominator

The denominator cannot be zero:

$$x-3\ne 0.$$

Therefore,

$$x\ne 3.$$

Step 3: Combine the restrictions

We need $x\ge -2$, but we must leave out $x=3$. Thus, the domain is

$$\boxed{[-2,3)\cup(3,\infty)}.$$

The bracket at $-2$ shows that $-2$ is included, because $\sqrt{0}$ is defined. The number $3$ is excluded because it would make the denominator equal to zero.