3.1 Functions, 3.1 Relations and Functions, 4.6 Rational Functions and Their Graphs, 5.4 Properties of Rational Functions

Finding Domain: Rational Function

Example:  Classify the function as a polynomial function, rational function, or root function, and then find the domain.  Write the domain interval notation and set builder notation.

h(x) = {x^2+8}/{x^2+x-20}

Solution:

Classify the Function

Formal Definition
Practical Way of Identifying
Polynomial Function

A polynomial function is a function of the form

f(x)=a_n x^n+a_{n-1} x^{n-1}+...+a_1 x+a_0

where n is a non-negative integer {0, 1, 2, 3, 4, …} and the coefficients a_n , a_{n-1,...,a_1,a_0 are from the real numbers.
Look for the variables to be in the numerator.  (If there is no fraction at all, they are in the numerator.)  The variable should not be inside a radical or absolute value.  The powers or exponents on the variables should be whole numbers.  Whole numbers come from this list {0, 1, 2, 3, 4, …}
Rational Function

A rational function is a function of the form

f(x)={p(x)}/{q(x)}

where p(x) and q(x) are polynomial functions and q(x) is not equal to zero.
The numerator and denominator are polynomials.  Most functions with variables in the denominator are considered Rational Functions but there are exceptions.
 Root Function (even index)

A root function is a function of the form

where n is an even positive integer greater than or equal to 2.

  The variable is inside or underneath a radical.  The index of the radical is an even number.  {2, 4, 6, 8, …}  The square root is an even index although the index is not written.
 Root Function (odd index)

A root function is a function of the form

where n is an odd positive integer greater than or equal to 2.

 The variable is inside or underneath a radical.  The index of the radical is an odd number.  {3, 5, 7, 9, …}  The cube root is an odd index.

Since the function h(x) = {x^2+8}/{x^2+x-20} has a variable in the denominator and the numerator and denominator are polynomial functions this function is a rational function.

Find the Domain of a Rational Function

Division by zero is undefined. Having a zero as the denominator is equivalent to division by zero thus is also undefined. The rational function is undefined for any value of the variable that gives a zero denominator. Find these values by creating an equation to solve. The equation is the expression in the denominator equal to zero.

Set the denominator equal to zero.
x^2+x-20=0
Solve the equation. This equation is a quadratic equation and can be solved by factoring, completing the square or the quadratic formula. x^2+x-20=0
Solve by factoring.  Factor the expression on one side. x^2+x-20=0
(x+5)(x-4)=0
Use the zero product property and set each factor equal to zero. (x+5)(x-4)=0
x+5=0 or x-4=0
Solve each equation. x+5=0 or x-4=0
x+5-5=0-5 or x-4+4=0+4
x=-5 or x=4

The values -5 and 4 give a zero value in the denominator, make the function undefined and must be excluded from the domain.

In set builder notation, the domain is delim{lbrace}{ x delim{ |} x!=-5 and x!=4}{rbrace}

In interval notation, the domain is (- infty,-5) union (-5,4) union (4,infty)