The Reciprocal Function

What is a Rational Function?

A function of the form $f\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}$, where $p(x)$ and $q(x)$ are polynomials, and $q\left(x\right)\ne0$, is called a rational function.

Some examples of rational functions are $f\left(x\right)=\frac{1}{x},\:f\left(x\right)=\frac{x-5}{x^2+2x-1}$, and $f\left(x\right)=\frac{3x^4-2x+5}{x^3-9x^2+1}$. 

Note About the Domain and Graph of a Rational Function:

Since any values of $x$ such that $q\left(x\right)=0$ are excluded from the domain of a rational function, this type of function often has a discontinuous graph - that is, a graph that has one or more breaks in it. Some values (those making $q\left(x\right)=0$) are excluded from a rational function's domain.

The Reciprocal Function

The simplest rational function with a variable denominator is the reciprocal function.

$f\left(x\right)=\frac{1}{x}$

The domain of this function is the set of all real numbers except 0. The number 0 cannot be used as a value of x, but it is helpful to find values of f(x) for some values of x very close to 0. We use the table feature of a graphing calculator to do this.

The tables suggest that  $\mid f\left(x\right)\mid$  increases without bound as x gets closer to 0, which is written in the symbols $\mid f\left(x\right)\mid\longrightarrow\infty$ as $\:x\longrightarrow0$.

The symbol $\:x\longrightarrow0$ means that x approaches zero, without necessarily ever being equal to 0. Since x cannot equal zero, the graph of $f\left(x\right)=\frac{1}{x}$ will never intersect the line $x=0$. This line is called a vertical asymptote.

As $\mid x\mid$ increases without bound, the values of $f\left(x\right)=\frac{1}{x}$ get closer and closer to 0, as can be shown from both positive and negative values using a calculator. An example shown for negative x values is below.

Letting $\mid x\mid$ increase without bound, written $\mid x\mid\longrightarrow\infty$, causes the graph of $f\left(x\right)=\frac{1}{x}$to move closer and closer to the horizontal line $\:y=0$. This line is a horizontal asymptote.

The Graph of the Reciprocal Function

Here are the domain and range of the function $f\left(x\right)=\frac{1}{x}$.

Domain: $\left(-\infty,0\right)\cup\left(0,\infty\right)$, and the Range: $\left(-\infty,0\right)\cup\left(0,\infty\right)$.

Let's make a T-chart to get some points for our graph.

x	y
$-2$	$-\frac{1}{2}$
$-1$	$-1$
$-\frac{1}{2}$	$-2$
0	Undefined
$\frac{1}{2}$	2
1	1
2	$\frac{1}{2}$

Here is what the graph of $f\left(x\right)=\frac{1}{x}$ looks like:

Characteristics of the Graph of the Reciprocal Function

• $f\left(x\right)=\frac{1}{x}$ decreases on the intervals $\left(-\infty,0\right)$ and $\left(0,\infty\right)$.
• It is discontinuous at $x=0.$
• The y-axis is a vertical asymptote, and the x-axis is a horizontal asymptote.
• It is an odd function, and its graph is symmetric with respect to the origin.