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

Answers Without Working