Rational Functions - Notesformsc

Rational functions are special functions that you cannot call polynomials, but are obtained by dividing polynomials. In other words, they are the quotients of the polynomial division.

A rational function is of form $f(x) = \frac{p(x)}{q(x)}$ where $p(x)$ and $q(x)$ are polynomials and

$q(x) \neq 0$ . The domain of rational function can be any real numbers except those that makes $q(x) = 0$ .

Example #1

$f(x) = \frac{x^2 - 9}{3x - 3}$ is a rational function. The function accept all real number except 1. The value 1 makes the function invalid. Hence, we can write domain of the $f(x)$ as $\{x \in R| x \neq 1\}$ .

Example #2

Find the domain of rational function $f(x) = \frac{2x}{x - 2}$ .

Solution:

The rational function accepts all values, except $2$ , which we can write in interval notation also, $(-\infty , 2) \cup (2, \infty)$ .

The Reciprocal Function

The simplest of rational function is the reciprocal function $f(x) = \frac{1}{x}$ . The function accepts all real values except $0$ .

Let us plot the graph of this rational function for following values.

X	-1	–0.5	-0.1	-0.01	-0.001	-0.0001
F(x)	-1	-2	-10	-100	-1000	-10000

As the $x$ value approaches $0$ from left, the value of $f(x)$ becomes smaller and smaller boundlessly to $-\infty$ . This can be shown with the arrow notation as follows.

Figure 1 - As x approaches 0 from left, the f(x) increases boundlessly to negative infinity — Figure 1 – As x approaches 0 from left, the f(x) decreases boundlessly to negative infinity

$x \to 0^{-} , f(x) \to -\infty$

$x \to 0^{-}$ , $f(x) \to -\infty$

What happens when $x$ comes closer to $0$ from right side.

X	0.0001	0.001	0.01	0.1	0.5	1
F(x)	10000	1000	100	10	2	1

Figure 2 - As x decreases and comes closer to 0, the f(x) becomes larger and moves towards positive infinity — Figure 2 – As x decreases and comes closer to 0, the f(x) becomes larger and moves towards positive infinity

$x \to 0^+$ , $f(x) \to \infty$ .

As the $x$ value approaches 0 from right , the value of $f(x)$ increases boundlessly to positive $infty$ . This is shown above with arrow notation.

What happens when the value of $x$ moves away from $0$ , that is, $x$ increases or decreases boundlessly ?

X	1	10	100
F(x)	1	0.1	0.01

x increases

X	-1	-10	-100
F(x)	-1	-0.1	-0.01

x decreases

Figure 3 – x increases or decreases boundlessly

When the $x$ value increases or decreases boundlessly, then the $f(x)$ approaches $0$ , but not touching $0$ .

This is shown in arrow notation below.

$x \to \infty$ , $f(x) \to 0$ and $x \to -\infty$ , $f(x) \to 0$ .

Figure 4 - Graph of reciprocal function — Figure 4 – Graph of reciprocal function

Vertical Asymptotes

There are several rational functions, out of which $f(x) = \frac{1}{x^2}$ is an interesting one. The graph of this function is reflected across the y-axis.

Figure 5 - Graph of f(x) = 1/x^2 — *Figure 5 – Graph of f(x) = 1/x^2*

We can describe the end behavior of this graph in the following manner.

The line $x = 0$ is called the vertical asymptote of the graph. A rational function can have

one vertical asymptote
many vertical asymptotes
or no vertical asymptotes

The end behavior of rational function around vertical asymptote are:

Figure 7 - As x approaches a f(x) decreases boundless

Figure 7 – As x approaches a, f(x) increases boundless

As the value of $x$ approaches the $a$ , the $f(x)$ increases or decreases without bound.

This increase or decrease in end behavior is useful in study of calculus. We can describe how the value of $x$ and the function changes using limits.

The $x$ moves closer to $a$ from left or right, the end behavior changes is shown in limits below.

How To Locate The Vertical Asymptotes ?

If the rational function have vertical asymptotes, then it can be found easily. We know that $f(x) = \frac{p(x)}{q(x)}$ where $p(x)$ and $q(x)$ are two polynomials. There are two conditions to find the vertical asymptotes:

The polynomials $p(x)$ and $q(x)$ have no common factors, if they have common factor must be eliminated.
The value $a$ must be zero of the polynomial $q(x)$ , that is, denominator. If $a$ is zero then $x = a$ is the vertical asymptotes.

You can understand this with the help of an example.

Example #3

Find the vertical asymptotes of the rational function: $f(x) = \frac{x^2}{x - 3}$

Solution:

The given equation $f(x) = \frac{x^2}{x - 3}$ does not have any common factors, therefore, meet the first condition. The denominator accepts all real numbers except $3$ which makes it $0$ .

Therefore, $a = 3$ is the vertical asymptote in the graph of rational function.

Example #4

Find the vertical asymptote of the rational function: $h(x) = \frac{(x + 2}{x^2 - 4}$ .

Solution:

In the given equation, $h(x) = \frac{(x + 2)}{(x - 2)(x + 2)}$ has a common factor. We reduce the common factor and the equation becomes $h(x) = \frac{1}{(x - 2)}$ .

The function accepts all real values, $(-\infty, 2) \cup (2,\infty)$ except $2$ . Therefore, $x = a = 2$ is the vertical asymptotes.

Example #5

Find the vertical asymptotes for the rational function: $p(x) = \frac{x - 6}{x^4 + 1}$ .

Solution:

The function has no common factor, but there is no $x$ value for which the denominator is $0$ . Therefore, the function does not have a vertical asymptote.

In some cases, the denominator is shows that it has a zero, but after reducing the common factors, the resultant expression has a totally new vertical asymptote.

Example #6

Find the vertical asymptote for the equation: $f(x) = \frac{(x - 1)(x + 3}{(x - 1)}$ .

Solution:

At first we see that the equation has a zero $x = a = 1$ , but when the equation is reduced after reducing the common factors, we get $x + 3$ and the vertical asymptote is $x = a = -3$ .

Horizontal Asymptotes

The equation $x = a$ represents “vertical asymptote“, similarly, $y = b$ represents the “horizontal asymptotes“. There may be several vertical asymptotes, but there is only one horizontal asymptote.

The function $f(x) = \frac{1}{x}$ , as $x$ increases or decreases without bound $x \to \infty$ or $x \to -\infty$ , the $f(x)$ approaches $0$ , which is $f(x) \to 0$ . So, we can say that the horizontal asymptote is that which is defined as $x \to \pm\infty, f(x) \to b$ .

Figure 8 - Horizontal Asymptote — Figure 8 – Horizontal Asymptote

We can write them in limit form as:

$\lim_{x \to a^{-}f(x) = \infty$	$\lim_{x \to a^{-}f(x) =- \infty$
$\lim_{x \to a^{+}f(x) = \infty$	$\lim_{x \to a^{+}f(x) = -\infty$