Graphs of functions can be used to determine the domain and range. The graphs give us an idea of which values of *x* and which values of *y* are being taken. Many times this can be enough to fully determine the domain and range. However, there will be times when we will have to use algebra to check or be more specific.

Here, we will learn how to determine the domain and range of a graph of a function. We will look at several examples to illustrate these ideas.

## How to find the domain and the range?

### Domain

We remember that the domain of a function is the set of all possible values of the independent variable. That is, the domain is the set of all values of *x* that will result in real values of *y*.

To find the domain, we must remember that:

- The denominator of a fraction cannot be equal to zero.
- The number under a square root cannot be a negative number.

We determine the domain by finding all the values of the independent variable that we can use and excluding the values that do not result in actual outputs of the function.

### Range

We remember that the range of a function is the set of all possible values of the dependent variable. That is, the range is the set of all values of *y* that result from using the domain as input.

The range can be determined by finding the maximum or minimum values of *y*. Also, we can substitute different values of *x* and analyze what kind of values of *y* we get. Graphs are particularly useful for determining the range.

## Examples of domain and range of a graph

### EXAMPLE 1

Find the domain and range of the function *f* that has the following graph:

**Solution:** We can see that the graph extends horizontally from -2 to 3, but the -2 is not included. Therefore, the domain is (-2, 3].

Also, we see that the graph extends vertically from 2 to -2, so the range is [-2, 2].

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### EXAMPLE 2

The following graph represents the function $latex f(x)= \frac{1}{x + 5}$. Determine its range and domain.

**Solution:** We can see that the graph extends horizontally beyond what we can see on the graph, so we can assume that it extends from negative infinity to positive infinity.

However, we can see that the graph never has a value of *x* at 5, so we have to exclude this value from the domain. Therefore, the domain is all real numbers of *x* and $latex x \neq -5$.

Similarly, we see that the graph extends vertically beyond what we can see in the graph, so we say that it extends from negative infinity to positive infinity. Here, we see that the y-values never reach zero, so we exclude them from the range.

Therefore, the range is all real values of $latex f(x)$ and $latex f(x \neq 0)$.

### EXAMPLE 3

The following graph represents the function f (x) = {{x} ^ 2} +5. Determine its range and domain.

**Solution:** This is a quadratic graph, so it stretches horizontally from negative infinity to positive infinity. That means that the domain is all real numbers of *x*.

We also see that the graph extends vertically from 5 to positive infinity. Therefore, the range is all real numbers of *y* and $latex y≥5$.

### EXAMPLE 4

The following is the graph of the function $latex \sqrt{-t+2}$. What is its domain and its range?

**Solution:** We see that the function extends indefinitely to the left and takes values up to 2. Then, its domain is all real numbers of *t*, where $latex t \leq 2$.

In the case of the range, we see that the function only takes positive values and extends upwards indefinitely. Therefore, the range is all real numbers of $latex f(t)$ and $latex f(t)≥0$.

### EXAMPLE 5

Find the domain and range of the following function:

**Solution:** We can see that this function has horizontal values that start at $latex x=-2$ and extend indefinitely to the right. However, we see that the function has an asymptote at $latex x=3$, that is, the function never takes that value.

Therefore, the domain of the function is all real numbers of *x*, where $latex x \geq -2$ and $latex x \neq 3$.

In the case of the range, we see that the graph extends from negative infinity to positive infinity. We also see that, although it seems that the graph never results in $latex y= 0$, we can see that the function takes this value at the point $latex x=-2$.

Therefore, the range of the function is all real numbers in *y*.

## See also

Interested in learning more about the domain and range of functions? Take a look at these pages:

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