RATIONAL FUNCTIONS: POLES, HOLES, DISCONTINUITIES, ASYMPTOTES, AND GRAPHS

Rational Functions
What are rational functions? Rational functions also known as algebraic fractions are functions that can be expressed as the quotient of two polynomials. Rational functions can be expressed in two major forms. The transformational form and the quotient form.

1. The Transformational Form: The transformational form of a rational function is given by:
\(\begin{array}{l}f\left( x \right) = \frac{a}{{x - h}} + k\\\begin{array}{*{20}{c}}{{\rm{where}}}&{h = {\rm{ }}\begin{array}{*{20}{c}}{{\rm{Horizontal}}}&{{\rm{Shift}}}\end{array}}\end{array}\\\begin{array}{*{20}{c}}{{\rm{and}}}&{k = {\rm{ }}\begin{array}{*{20}{c}}{{\rm{Vertical}}}&{{\rm{Shift}}}\end{array}}\end{array}\end{array}\)

2. The quotient Form: The quotient form as the name suggests is simply expressing the rational as the quotient of two polynomial function. This form is given by:
\(\begin{array}{l}f\left( x \right) = \frac{{N\left( x \right)}}{{D\left( x \right)}}\\{\rm{where}}\\\begin{array}{*{20}{c}}{N\left( x \right) = }&{{\rm{The}}}&{{\rm{Numerator}}}&{{\rm{Function}}}&{}\\{D\left( x \right) = }&{{\rm{The}}}&{{\rm{Denominator}}}&{{\rm{Function}}}&{}\end{array}\end{array}\)
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Examples of Rational functions are as follows
​\(\begin{array}{l}f\left( x \right) = 3{x^2} + 3x = \frac{{3{x^2} + 3x}}{1}\\f\left( x \right) = \frac{{2\left( {x - 2} \right)}}{{\left( {x + 2} \right)}}\\f\left( x \right) = \frac{4}{{{x^2} - 3x - 4}}\end{array}\)

ANALYSIS OF RATIONAL FUNCTIONS

Domains
The domain of a rational function is the set of all real numbers except the real numbers that cause the numbers to attain a value of 0. The domain restrictions can be determined by computing the roots of the denominator polynomial function.
In symbolic form the domain of \[f\left( x \right) = \frac{{N\left( x \right)}}{{D\left( x \right)}}\] is given by the set of all real numbers where \[D\left( x \right) \ne 0\].

Consider the example below


Example 1:

Find the domain of \(f\left( x \right) = \frac{{3\left( {x - 2} \right)}}{{\left( {x - 3} \right)\left( {x + 1} \right)}}\).


Steps for finding the domain:
1. Extract the denominator functions
2. Find the roots by setting equal to zero and solving for x. 

Solution to Example 1:
1. The denominator function 
\(D\left( x \right) = \left( {x - 3} \right)\left( {x + 1} \right)\)
2. Find the roots by solving D(x)=0
\(\left( {x - 3} \right)\left( {x + 1} \right) = 0\)

\(x = 3,\, - 1\)

These values will have to be excluded from the set of reals to get the domain of f(x)
Domain = All real numbers except 
\(x = 3,\, - 1\) or in interval notation domain=\(\left( { - \infty  - 1} \right) \cup \left( { - 1,3} \right) \cup \left( {3,\infty } \right)\). Using set notation the domain can be expressed as \(\mathbb{R} - \left\{ { - 1,3} \right\}\)                                                  

Rational Functions Playlist

Finding The Vertical Asymptotes (How to):
Let f  be a rational function where \(f\left( x \right) = \frac{{N\left( x \right)}}{{D\left( x \right)}}\). The vertical asymptotes can be found by following the given steps:

1. Factor the numerator and denominator completely
2. Reduce by cancelling out all common factors 
3. Set the remaining part of the denominator to zero and solve.

Note: There are cases where you have no vertical asymptotes.

Example 2: Find the vertical asymptotes of the rational function\(y = \frac{{{x^2} - 4}}{{3{x^3} - 5{x^2} - 2x}}\)
1. (Factor) 
\(y = \frac{{{x^2} - 4}}{{3{x^3} - 5{x^2} - 2x}}\)
\(\,\,\,\, = \frac{{\left( {x - 2} \right)\left( {x + 2} \right)}}{{x(x - 2)(3x + 1)}}\)
2. (Cancel out all common factors)\(\,\,\,\, = \frac{{\left( {x + 2} \right)}}{{x(3x + 1)}}\) =\(\frac{{x + 2}}{{x\left( {3x + 1} \right)}}\)
3.  (Find the zeros of the remaining denominator function i.e. set=0 & solve)
\(x\left( {3x + 1} \right) = 0\)
\(x = 0,\,\,\,x =  - \frac{1}{3}\)
The vertical asymptotes are the lines \(x = 0,\,\,\,x =  - \frac{1}{3}\)

HOW TO SOLVE RATIONAL EQUATIONS
1.Rewrite terms with their denominators equal to their LCD
2.Eliminate all denominators by multiplying by the LCD
3.Solve and Check​