Domain and Range – Definition and Examples

EXAMPLE

The following is the graph of $latex y=\sqrt{{x + 2}}$:

The domain of this function is $latex x\ge -2$ because x cannot be less than -2. To verify this, we can try the number -3. Replacing $latex x=-3$, we have $latex y=\sqrt{{- 3+2}} = \sqrt{{- 1}}$.

We have a negative number within a square root and the result is not a real number. This means that only values of x greater than or equal to -2 produce real values in the function.

EXAMPLE

Again, let’s look at the graph of $latex y=\sqrt{{x+2}}$:

We can see that the curve is always above the horizontal axis. No matter what value of x we try, we will always get a value of y that is zero or positive. In this case, the range is $latex y \ge 0$.

The graph goes to the right indefinitely, so the range is all non-negative values of y.

EXAMPLE

Find the domain and range of the function $latex f (x)=\frac{\sqrt{x+2}}{{{x}^2}-9}$ without using a graph.

Solution: In the numerator of the fraction, we have a square root. To ensure that the value under the root is non-negative, we can only use values ​​of x that are greater than or equal to -2.

The denominator of the fraction has the expression $latex {{x}^2}-9$, which can be written as $latex (x + 3)(x-3)$. Therefore, our values ​​for x cannot include -3 for the first parenthesis and 3 for the second parenthesis.

Thus, the domain for this function is $latex x \ge -2, ~~ x\ne 3$.

To determine the range, we consider the numerator and denominator of the function separately:

Numerator: If $latex x=-2$, we have $latex f (x)=\sqrt{{- 2 + 2}}=0$. As x increases, the numerator will also increase to infinity.

Denominator: We divide into the following parts:

When $latex x=-2$, the lower part of the function equals $latex {{(-2)}^2}-9=4-9=-5$. Then, we have $latex f(-2)=\frac{0}{- 5} = 0$.

Between $latex x=-2$ and $latex x= 3$, the expression $latex {{(x)}^2} – 9$ approaches 0, so the function will tend to infinity as we approach $latex x=3$.

For $latex x>3$, when x is slightly larger than 3, the denominator value is slightly larger than 0, so the function will have a very large positive value.

For very large values ​​of x, the numerator will be large, but the denominator will be much larger, so the value of the function will get smaller and smaller.

Therefore, the range of the function is $latex (-\infty, ~ 0\big], ~ (\infty, ~ 0)$.

We can visualize this in the graph:

EXAMPLE 1

Find the domain and range for the function $latex f(x)={{x}^2}+5$.

Domain: The function $latex f(x)={{x}^2}+5$ is defined for all values of x since there is no restriction on the value of x. Therefore, the domain of $latex f(x)$ is “all real values of x“.

Range: Since $latex {{x}^2}$ is never negative, the function $latex f(x)={{x}^2}+5$ is never less than 5. Therefore, the range of $latex f(x)$ is “all real numbers $latex f(x) \ge 5$.

In the graph, we can see that the domain is all real numbers, but the range is all real numbers greater than 5:

EXAMPLE 2

Find the domain and range for the function $latex f(x)=\frac{1}{x+5}$.

Domain: The function $latex f(x)=\frac{1}{x+5}$ is not defined for $latex x=-5$ since this value would produce a division by 0. Therefore, the domain of the function is all real numbers with the exception of -5.

Range: No matter how big or how small the values of x are, the function $latex f(x)$ will never equal 0. This means that the range of the function is all real numbers except 0.

In the graph we can see that the function is not defined for $latex x=-5$ and that the range is all integers except 0:

EXAMPLE 3

Find the domain and range for the function $latex f(t)=\sqrt{{2-t}}$.

Domain: The function $latex f(t)=\sqrt{{2-t}}$ is not defined for values greater than 2 since it would cause us to have a negative value within the square root. Then, the domain of the function is all real numbers less than or equal to 2, $latex t\le 2$.

Range: By definition we have $latex f(t)=\sqrt{{2-t}} \ge 0$.

In the graph we can see that t does not take values greater than 2 and the range is greater than or equal to 0: