Rational Function

A Rational Function is a type of function that is expressed as a fraction, where both the numerator and denominator are polynomial, and the denominator cannot be equal to zero. In simple words, a rational function can be defined as the ratio of two polynomials.

Rational Function is a mathematical expression that represents the ratio of two polynomial functions. The key condtion is that the denominator must never be zero.

Mathematical Representation:

A rational function can be written as: \mathbf{f(x)} = \frac{\mathbf{g(x)}}{\mathbf{h(x)}}
Where f(x), g(x) and h(x) are all polynomials in variable x and h(x) ≠ 0.

Rational Functions Examples

Some examples of rational functions are as follows:

• f(x) = x + 1/(x + 2),
• f(x) = x² - 1/(x² + 1),
• f(x) = 3x/(x² - 4),
• f(x) = x³/(x + 8),
• f(x) = 2x/(1 - 4x),
• f(x) = 7x²/(8x³ + 4).

Note:- A function where the numerator is a polynomial and the denominator is a constant (other than zero) is called a linear function, not a rational function.

Properties of Rational Function

A rational function has various properties which are as follows:

• Domain of Rational Function
• Range of Rational Function
• Asymptotes of Rational Function
  • Horizontal Asymptote
  • Vertical Asymptote
  • Oblique Asymptote
• Holes of Rational Function

Domain of Rational Function

We know that the denominator is never zero in a rational function. This fact is used to find out the domain and range of the rational function. To calculate the domain of the rational function following steps are followed

• Keep the value of the denominator equal to zero
• Find the values of the variable where the denominator becomes zero.
• The domain of the function is thus the set of all real numbers R excluding the values which make the denominator zero.

Range of Rational Function

To calculate the range following steps are followed:

• Set f(x) = y
• Solve the equation so obtained for the variable x.
• Now use the condition of denominator nit equals to zero.
• Find the value of y from the above condition.
• The range of the function is a set of all real numbers R excluding the value of y so obtained.

Let's consider an example for understanding the domain and range of Rational Functions.

Example: Consider a rational function (x+2)/(x+1), and find its domain and range.

Solution:

To calculate the domain of the given function, we can follow the following steps:

Equate the denominator to zero i.e. x + 1 = 0, which gives x = -1.

Thus the domain of the function is R - {-1}.

The range of the function is calculated as follows:

Set f(x) = y = (x + 2)/(x + 1)

Solve the equation for x

y(x + 1) = x + 2
⇒ xy + y = x + 2
⇒ xy - x = 2 - y
⇒ x(y - 1) = 2 - y
⇒ x = 2 - y/(y - 1)

Keep denominator not equals to zero i.e. y - 1 ≠ 0 which gives y ≠ 1.

Thus range of the function is set of real numbers R - {1}.

Asymptotes of Rational Function

All the Rational Functions have three types of Asymptotes and they are,

• Vertical Asymptote
• Horizontal Asymptote
• Oblique Asymptote

Now, let's learn about them in detail

Vertical Asymptote

A vertical asymptote of a rational function is a line parallel to the y-axis and is of the form x=a where a is any number. This line appears to touch the graph of the rational function but it never touches it. There can be one or more vertical asymptotes of a rational function. To find the vertical asymptote of the rational function, the following steps are followed:

• Reduce the rational function to its lowest form and eliminate all the common terms in numerator and denominator.
• Set the denominator equal to zero and find the value of the variable for which the denominator becomes zero.

Horizontal Asymptote

A horizontal asymptote of a rational function is a line parallel to the x-axis and is of the form y=a where a is any number. This line appears to touch the graph of the rational function but it never touches it. There can be only one horizontal asymptote of a rational function. The horizontal asymptote of a rational function can be calculated by comparing the degrees of the numerator and denominator as follows:

• Let N and D be the degree of the numerator and denominator respectively.
• If N < D, the horizontal asymptote is y=0
• If N > D, then there is no horizontal asymptote
• If N = D, y = ratio of leading coefficients in numerator and denominator is the horizontal asymptote.

Oblique Asymptote

It is a slant line which appears to touch the graph of the rational function but never touches it. It is present only in the case where degree of numerator N = degree of denominator D+1. The horizontal asymptote is equal to the quotient which is the result of the division of numerator and denominator of the rational function.

Holes of Rational Function

The points that appear to be present on the graph of the rational function but are not present are called holes of a rational function. To calculate the holes, first reduce the function to the lowest form and set the common factor equal to zero. The value of the variable and the value of the function at that variable is the required hole.

Check - Graph a Rational Function with Holes

Simplifying Rational Functions

To simplify the rational function, we need to reduce it to its lowest form. The steps to be followed are:

• Step 1: Factorise the numerator and denominator.
• Step 2: Cancel out the common terms from both the numerator and denominator. The function thus obtained is the simplified form of the given rational function.

For example: Consider the function (x² + 2x + 1)/(x² + 3x + 2).

Step 1: Factorise the numerator and denominator.

• Numerator: x² + x + x + 1
  This factor as: (x + 1) (x + 1)

• Denominator: (x² + x + 2x + 2)
  This factor as: (x + 1) (x + 2)

So the function becomes:
(x + 1)(x + 1)/(x + 1)(x + 2)

Step 2: Cancel out common terms

The term (x + 1) appears in both the numerator and the denominator, so we can cancel it out

Thus the function is simplified and written as: x+1/(x+2)

Operations On Rational Functions

All the normal operations in mathematics can also be applied to rational functions. These basic operations are addition, subtraction, multiplication, and division. We will see how to perform these functions by considering two rational functions a/b and c/d.

Addition of Rational Functions

The addition of any two rational functions a/b and c/d is done as:

\frac{a}{b}+\frac{c}{d} = \frac{ad+bc}{bd}

Subtraction of Rational Functions

Subtraction of any two rational functions a/b and c/d is done as:

\frac{a}{b}-\frac{c}{d} = \frac{ad-bc}{bd}

Multiplication of Rational Functions

Multiplication of any two rational functions a/b and c/d is done as:

\frac{a}{b}\times \frac{c}{d} = \frac{ac}{bd}

Division of Rational Functions

Division of any two rational functions a/b and c/d is done as:

\frac{a}{b}\div \frac{c}{d} = \frac{ad}{bc}

Graphing Rational Functions

To graph a rational function we need to follow the following steps:

Step 1: Find the vertical asymptote of the given rational function and mark it with a dotted line on the graph.

Step 2: Find the vertical asymptote of the given rational function and mark it with a dotted line on the graph.

Step 3: Find the holes of the rational function if they exist.

Step 4: Calculate the x and y-intercept of the function by setting y and x equal to zero alternately.

Step 5: Draw a table containing two columns labelled x and y. Now write the x-intercepts and vertical asymptotes of the rational function in this table. Choose random numbers and write them in the column labelled x on either side of each of the x-intercepts and vertical asymptotes.

Step 6: Calculate the value of y or f(x) by substituting the chosen value of x in the rational function.

Step 7: Plot all the points obtained in the table and join them with a free hand as if drawing a curve without touching the asymptotes.

Example:Graph the function f(x) = (x + 3) / (x - 1).

Solution:

Given f(x) = (x + 3) / (x - 1) = y(let)

Calculate the vertical asymptote by setting the denominator equal to zero:
x -1 = 0
x = 1

Calculate horizontal asymptote.
As degree of numerator and degree of denominator is same, horizontal asymptote is ratio of leading coefficients i.e. 1:1 which is y = 1.
As there is no common factor between numerator and denominator, holes do not exist.

Find x intercept by putting y = 0.
Thus x intercept is:
(x + 3) / (x - 1) = 0
x = -3

So x intercept is (-3,0).

Find y intercept by putting x = 0.
3/(-1) = y
y = -3

So y intercept is (0, -3).

Draw a table labelled x and y.
Now write the x-intercepts and vertical asymptotes of the rational function in this table. Choose random numbers and write them in the column labelled x on either side of each of the x-intercepts and vertical asymptotes as follows:

x	-5	-4	-3	-2	0	1	2	3
y	(-5+3)/(-5-1) = 1/3 = 0.3	(-4 + 3) / (-4 - 1) = 1/5 = 0.2	0 (x intercept)	(-2 + 3) / (-2 - 1) = -1/3 =-0.33	-3 (y -intercept)	Vertical symptote	5	3

Plot all these points on the graph as follows:

Read More,

• Domain and Range of a Function
• Relation and Function

Solved Examples Rational Functions

Example 1: Find the vertical asymptote of (x+1)/(2x+3).

Solution:

Given f(x) = (x+1)/(2x+3)

As the function is already in its lowest form and there is no common factor between numerator and denominator, vertical asymptote is calculated by simply equating the value of denominator equal to zero.
2x + 3 = 0
x = -3/2

Thus, x = -3/2 is the vertical asymptote of the given function.

Example 2: Find the horizontal asymptote of x³+x²+1/(2x+1).

Solution:

Given f(x) = x³+x²+1/(2x+1)

Degree of numerator N = 3
Degree of denominator D = 1

As N>D, there is no horizontal asymptote for given function.

Example 3: Find the oblique asymptote of (2x²+3)/(x+4).

Solution:

Given f(x) = 2x²+3/(x+4)

As degree of numerator = 2 = degree of denominator + 1, oblique asymptote is the quotient that is obtained by dividing numerator and denominator. The division is shown below:

As the quotient obtained is 2x, it is the oblique asymptote to the given function.

Example 4: Find the domain of the function x/(x²-1).

Solution:

Given f(x) = x/(x²-1)
Equate the denominator to zero.

x² - 1 = 0
x = +1 or -1

Thus domain of the given function is set of real numbers excluding 1 and -1 which is R - {-1, 1}.

Example 5: Find the range of the function 4x+5/(6x+7).

Solution:

Given f(x) = 4x+5/(6x+7)

Set f(x) = y

y = 4x+5/(6x+7)

Solve for x

y(4x + 5) = 6x +7
4xy + 5y = 6x + 7
4xy - 6x = 7 - 5y
x(4y - 6) = 7 - 5y
x = 7 - 5y/(4y - 6)

Keep denominator not equal to zero

4y - 6 ≠ 0
y ≠ 6/4
y ≠ 3/2

Thus the range of function is set of real numbers excluding 3/2 which is R - {3/2}

Practise Problems on Rational Functions

Problem 1: Simplify the rational expression: \frac{2x^2 - 8}{4x^2 - 16}

Problem 2: Find the domain of the rational function: f(x) = \frac{x^2 - 9}{x - 3}

Problem 3: Determine the vertical asymptotes, if any, for the rational function: h(x) = \frac{x^2 - 4x - 5}{x^2 - 5x + 6}

Problem 4: Solve for x in the equation: \frac{3x - 1}{x + 2} = 2

Problem 5: Given the rational functiong(x) = \frac{2x^3 - x^2 - 6x}{x^3 - 3x^2 - 4x} , find the horizontal asymptotes, if they exist.