Rational Functions - Notesformsc

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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.

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.

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:

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

The Reciprocal Function

Vertical Asymptotes

How To Locate The Vertical Asymptotes ?

Horizontal Asymptotes

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