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.
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:
\[\text{Domain} = \left \{ x \in \mathbb{R} | x \neq -2 \right \}\]
To find a function's domain we should ask ourselves:
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.
Find the domain of each of the following functions: