Define Rational Function

Introduction to define rational function:

Rational numbers is define as an integer that holds the value which is not equal to 0 and the value form of `x/y` , where x, y is two integers and` y != 0` . Get-together of each and every rational integer(numbers) is denoted by y. A rational integer(numbers) form is said to be proper form, when x and y having no similar factor other than 1. All integers(numbers) are said to be rational numbers. Rational function is referred as f(x). Let us discuss about the rational function in breif.

Define Rational Function:
Rational function is defined as a form that is,

`f(x) = ((p(x))/(q(x)))`

A rational function is generally a division of two polynomial functions mentioned as p(x) and q(x). That is, it is a polynomial divided by an another polynomial. Thus the rational function is defined with the definition and the form of the function is shown. Now let we have some example problems for clear understanding.

Example - Define Rational Function:

Example - Define rational function 1:

Determine the domain for the given rational function f(x) = `(x - 5)/(x^2 + 4x - 12)`

Solution:

Step 1: Given rational function is,

f(x) = `(x - 5)/(x^2 + 4x - 12)`

Step 2: Set the denominator of the given function f(x) = `(x - 5)/(x^2 + 4x - 12)` to zero and solve for x

x2 + 4x - 12 = 0

By rewriting the above equation,

x2 + 6x - 2x - 12 = 0

By separating common terms, we get

x(x + 6) - 2(x + 6) = 0

(x - 2)(x + 6) =0

By factoring each term, we get

x - 2 = 0

x = 2

x + 6 = 0

x = -6

Thus, the domain value of the given function is all real numbers except  for x, x = 2 or - 6

Example - Define rational function 2:

Define the domain of the function f(x) = `(7x)/(18x - 9)`

Solution:

Step 1: Given rational function is,

f(x) = `(7x)/(18x - 9)`

Step 2: Set the denominator of the given function f(x) = `(7x)/(18x - 9)` to zero and solve for x

18x - 9 = 0

By adding 9 on both sides, we get

18x =  9

Divide 18 on both sides, therefore

x = `9/18 `

x = `1/2`

x = 0.5

Therefore, the domain of a given rational function is all real numbers except for x = 0.5.