# Domain & Range 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.

#### Example 2

Consider the function $$f(x)=x^2-1$$ for the following values of $$x$$: $-3,-2, 0, 1, 2$ \begin{aligned} f(-3) &= (-3)^2 - 1 \\ & = 9 - 1 \\ f(-3) &= 8 \end{aligned}

Calculating all of the other values of $$x$$ this way we find:

• For the input value $$x = -2$$, the output value is: \begin{aligned} f(-2) &= (-2)^2 - 1 \\ & = 4 - 1 \\ f(-2) & = 3 \end{aligned} So $$-2$$ maps onto $$3$$
• For the input value $$x = 0$$, the output value is: \begin{aligned} f(0) &= 0^2 - 1 \\ & = 0 - 1 \\ f(0) & = -1 \end{aligned} So $$0$$ maps onto $$-1$$

• For the input value $$x = 1$$, the output value is: \begin{aligned} f(1) &= 1^2 - 1 \\ & = 1 - 1 \\ f(1) & = 0 \end{aligned} So $$1$$ maps onto $$0$$
• For the input value $$x = 2$$, the output value is: \begin{aligned} f(2) &= 2^2 - 1 \\ & = 4 - 1 \\ f(2) & = 3 \end{aligned} So $$2$$ maps onto $$3$$

Can be illustrated in a mapping diagram, as shown here:

## Definition - Domain of a Function

Given a function, its domain is the set, or interval, of numbers for which the function is well defined and can be calculated.
For the function we saw in the introduction, $$f(x) = \frac{2}{3x+6}$$, 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.

## Must Know Examples

### Example 1

Find the domain of the function defined as: $f(x) = \frac{3}{2x-4}$

### Example 2

Find the domain of the function defined as: $f(x) = 2\sqrt{x+3}$

### Example 3

Find the domain of the function defined as: $f(x) = \frac{3}{x^2-4}$

### Example 4

Find the domain of the function defined as: $f(x) = \frac{3}{\sqrt{2x+8}}$

## 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.

## Reading Domain & Range from Graphs

Given a function $$f(x)$$ and its curve $$y=f(x)$$, we can read both the domain and range as follows:

• The domain is the interval of $$x$$ values seen when we project the curve on to the $$x$$-axis.
• The range is the interval of $$y$$ values seen when we project the curve on to the $$y$$-axis.
Each of these are illustrated in the example below.

## Example

The following graph shows the curve of some function $$f(x)$$:

Using the graph, state:

1. the domain
2. the range.

### Solution

1. To find the domain, we look at the interval of $$x$$ values seen by projecting the curve onto the $$x$$-axis; this is done here in blue:
We can see that the curve projects onto the interval: $-6\leq x \leq 20$ More formally, we can now state that the domain is: $\text{Domain} = \left \{ x \in \mathbb{R}| -6\leq x \leq 20 \right \}$
2. To find the range, we look at the interval of $$y$$ values seen by projecting the curve onto the $$y$$-axis; this is done here in green:

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