# Domain of a Function

Given a function, $$f(x)$$, there may be some values of $$x$$ for which the function isn't well defined.
For instance, if we were asked to calculate the value of the function $$f(x) = \frac{2}{3x+6}$$ when $$x = -2$$ we would have some trouble.
Indeed, if we replace $$x$$ by $$-2$$ in the expression for $$f(x)$$ we obtain: \begin{aligned} f(-2) & = \frac{2}{3\times (-2) + 6} \\ & = \frac{2}{-6+6} \\ f(-2) & = \frac{2}{0} \end{aligned} Since it is impossible to divide by zero, it is impossible to calculate the value of $$f(x)$$ when $$x = -2$$.
This highlights the importance of defining the domain of a function.

## Definition - Domain of a Function

Given a function, its domain is the set, or interval, of numbers for which the function is well defined.
For the function we saw in the intriduction, $$f(x) = \frac{2}{3x+6}$$, the domain would be all real numbers other than $$-2$$.
We write this: $\text{Domain} = \left \{ x \in \mathbb{R} | x \neq -2 \right \}$

### How to Find the Domain

To find a function's domain we should ask ourselves:

Are there any values of $$x$$ at which this function can't be calculated (isn't well defined)?

To answer this question we must always remember:

• Denominators cannot equal $$0$$.

• Expressions inside square roots (radicands) cannot be negative.
Once this question is answered then we can take the values of $$x$$, at which the function isn't well defined, away from the set of real numbers $$\mathbb{R}$$ to define the domain.

## Tutorial

In the following tutorial we review the method for findng the formula for the $$n^{\text{th}}$$ term of a linear sequence. Watch it now.

## Exercise

Find the domain of each of the following functions:

1. $$f(x) = \frac{3}{2x-4}$$

2. $$f(x) = \sqrt{x+5}$$

3. $$f(x) = \sqrt{3x-9}$$

4. $$f(x) = \frac{7}{9-x}$$

5. $$f(x) = \frac{x}{x^2-4}$$

6. $$f(x) = \frac{2x}{x^2+4}$$