Domain and Range of Rational Functions

The domain of a function $f\left(x\right)$ is all of the x values for which the function is defined, and the range is the set of all values that f takes. We've worked with both of these terms before, but it's time to expand them to all rational functions. In this article, we'll explore how to do that. Let's get started!

Finding the domain of rational functions

A rational function may be defined as any function in the form $f\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}$ , where $p\left(x\right)$ and $q\left(x\right)$ are both polynomials and $q\left(x\right)\ne 0$ .

The domain of a rational function is all real numbers x except those that would set the denominator to zero. We can find the exclusions by setting the denominator equal to zero and solving for x.

For instance, the domain of the parent function $f\left(x\right)=\frac{1}{x}$ is all real numbers except $x=0$ . Likewise, the domain of $f\left(x\right)=\frac{1}{x-4}$ is all real numbers except $x=4$ .

Let's consider a slightly more complicated example. The function $f\left(x\right)=\frac{\left(x+1\right)\left(x-2\right)}{\left(x-2\right)}$ can be simplified to the linear function $f\left(x\right)=x+1$ when $x\ne 2$ , but we still need to indicate that the function is not defined when $x-2$ . We do that by graphing a hole when $x=2$ as shown:

Finding the range of rational functions

There are different ways to find the range of rational functions, depending on the specific function. One common way is to graph the function and analyze its behavior. For some functions, finding the inverse and determining the domain of the inverse might help to find the range, but this method is not always applicable or straightforward.

Returning to the example of $f\left(x\right)=\frac{1}{x}$ , we know that $f\left(x\right)$ approaches infinity as x gets closer to zero from either side of it on the Cartesian plane. Here is what it looks like once graphed:

The graph approaches the x-axis as x gets closer to positive and negative infinity, but it never actually touches the x-axis. The function can therefore take all real values except 0. That means the range is all real numbers save 0.

The graph of $f\left(x\right)$ will never touch the x-axis or y-axis, and mathematicians have a word for lines like that: asymptote. Put simply, an asymptote is a line that the graph of a function approaches but never touches. We can find the vertical asymptote of a rational function by equating the denominator to zero and solving for x. If the degree of the polynomial in the numerator is less than that of the denominator, its horizontal asymptote will be the x-axis, or $y=0$ .

In more general terms, we can say that rational functions can be expressed in the form $f\left(x\right)=\frac{ax+b}{cx+d}$ . In this form, we can calculate the vertical asymptote as $x=-\frac{d}{c}$ . If the degrees of the numerator and denominator are the same, the horizontal asymptote is $y=\frac{a}{c}$ . If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $y=0$ .

Practice questions on the domain and range of rational functions

a. Find the domain and range of the following function: $y=\left(\frac{1}{x}+3\right)-5$

To find the excluded value in the domain of this function, set the denominator equal to 0 and solve for x:

$x+3=0$

$x=-3$

The domain is all real numbers except -3.

To find the range, we can analyze the behavior of the function as x approaches the excluded value and as x goes to positive or negative infinity:

As x approaches -3 from the left or right, the term $\frac{1}{x+3}$ goes to positive or negative infinity, respectively. Therefore, y can take on any value except -5, as the function will never actually equal -5 due to the subtraction.

As x goes to positive or negative infinity, the term $\frac{1}{x+3}$ approaches 0. This means y approaches -5.

We've determined that the domain of the function is all real numbers except -3 and the range is all real numbers except -5. We can express this mathematically as:

Domain: $\left\{x\in R | x\ne -3\right\}$

Range: $\left\{y\in R | y\ne -5\right\}$

Here is what it looks like graphically:

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