Rational Functions Learning

A function is in the form of  y = (f(x))/(g(x)) , where both f(x) and g(x) are polynomial functions is called rational function.

Examples of rational functions:    y = 1/(x - 2)

f(x) = (3x)/(4 + 5)

g(x) = (x^2 - 6)/(x^2 - 3x)

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These are the some of the examples of rational functions. The domain and different asymptotes of rational function are explained below in detail.

Learning domain of the rational functions:

In a function, domain is defined as the set of all possible real numbers of the independent variable i.e., it is defined as the set of all real numbers for which the function is defined. A example for learning domain of a rational function is given below.

Example:

1) Find domain of the function f(x) = (4x + 9)/(2x - 4)

Solution:

Step 1: Given function

f(x) = (4x + 9)/(2x - 4)

Step 2: Set denominator to zero.

2x - 4 = 0

Step 3: Solve the above equation for x.

2x = 4

Divide by 2 on both side,

x =2

Therefore, the domain of the given function is all real numbers except x = 2

Learning of asymptote of rational functions:

Asymptotes of rational functions are classified into three types as follows,

Horizontal asymptote
Vertical asymptote
Oblique asymptote

Horizontal asymptote:

In the graph of a rational function, the horizontal line approaches as ' x '  values get very large or very small. It is a line of the form y = c. If  value of x gets increase in positive or negative direction, the function f(x) will increase to the number c.

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Vertical asymptote:

In the graph of a rational function, the vertical line approaches as ' x ' values approach a fixed number. In vertical asymptote, the function f(x) becomes infinite. The function have vertical asymptote at x = a.

Oblique asymptote:

The oblique asymptote is in the form y = ax + b with non-zero. In the rational function, if the numerator has the highest degree than the denominator, then the rational function has oblique asymptote.

Learning about different asymptote helps you to draw graph of a rational function.