Rational Functions

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.

X0.00010.0010.010.10.51
F(x)1000010001001021
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 ?

X110100
F(x)10.10.01
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.

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

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

  1. one vertical asymptote
  2. many vertical asymptotes
  3. 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

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

\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

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:

  1. The polynomials p(x) and q(x) have no common factors, if they have common factor must be eliminated.
  2. 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 -\infty}f(x) = b lim_{x \to +\infty}f(x) = b