The domain of exponential functions is equal to all real numbers since we have no restrictions with the values that *x* can take. The range of exponential functions is equal to the values above or below the horizontal asymptote.

Here, we will see in detail how to find the domain and range of exponential functions. Also, we will look at several examples with the graphs of the functions to illustrate these ideas.

## How to find the domain and the range of exponential functions?

Recall that the domain is the set of input values that are used for the independent variable.

Also, remember that the range is the set of all the output values for the dependent variable.

For any exponential function with the general form $latex f(x)=a{{b}^x}$, the domain is the set of all real numbers. That is, we have:

$latex – \infty < x < \infty$

For any exponential function with the general form $latex f(x)=a{{b}^x}$, the range is the set of all real numbers above or below the horizontal asymptote, $latex y=d$. The range does not include the value of the asymptote, *d*. That is, we have:

If $latex a>0$, $latex f(x)> d$

If $latex a <0$, $latex f(x)<d$

## Examples of domain and range of exponential functions

### EXAMPLE 1

A simple exponential function like $latex f(x)={{2}^x}$ has a domain equal to all real numbers. However, its range is equal to only positive numbers, where, $latex y>0$. That is, the function $latex f(x)$ never takes a negative value.

Also, the function never reaches the value of 0 even though it gets very close as *x* approaches negative infinity.

### EXAMPLE 2

We can replace *x* with –*x* in the function in the previous example to obtain the function $latex g(x)={{2}^{- x}}$. We see that the graph was reflected with respect to the *y*-axis.

However, the domain and range do not change. The domain is equal to all real numbers. And the range is equal to only positive numbers, where, $latex y>0$.

### EXAMPLE 3

If we can now put a negative sign in front of the function, we have $latex h(x)=-{{2}^x}$. In this case, the graph is reflected with respect to the *x*-axis. The domain is still all real numbers of *x*. However, the range is now all negative numbers, where, $latex y<0$.

### EXAMPLE 4

Find the domain and range of the function $latex f(x)={{2}^{x+3}}$.

**Solution:** The graph of this function is simply the graph of $latex f(x)={{2}^x}$ translated 3 units to the left.

The function is defined for all real numbers, so the domain of the function is the set of real numbers.

As *x* approaches infinity, the function also approaches infinity, and as *x* approaches negative infinity, the function approaches the *x*-axis, but never touches it.

Therefore, the range is the set of all real numbers $latex \{y \in R | y> 0 \}$.

### EXAMPLE 5

What is the domain and range of the function $latex f(x)=1.5({{2}^x})+3$?

**Solution:** This function is also defined for all real numbers. Therefore, the domain of the function is the set of real numbers.

We see that in this case, $latex d=3$. This means that the horizontal asymptote is equal to $latex y=3$ and the function has a range that is equal to all real numbers greater than 3.

Therefore, the range is the set of all real numbers $latex \{y \in R | y> 3 \}$.

### EXAMPLE 6

What is the domain and range of the function $latex f(x)=-{{3}^x} – 1$?

**Solution:** The function can take any value of *x* as input. This means that the domain is the set of all real numbers.

The function has only negative values of *y*. Also, we see that we have $latex d=-1$, so the horizontal asymptote is equal to $latex y=-1$. Therefore, the range is all values less than -1.

The range is the set of all real numbers $latex \{y \in R | y <-1 \}$.

## See also

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

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