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Rational functions are functions defined by fractions of polynomials.
What are rational functions?
In algebra, polynomials play the same role as the integers do in arithmetic. We add
them, subtract them, multiply them, and factor them. We cannot divide them,
however, if we want an integer answer. Since is not a factor of , is not
an integer. If we want to be able to divide integers, we have to move to
the rational numbers, which are fractions where and are integers, and
The same idea holds for polynomials. We can add them, subtract them, multiply
them, and factor them. However, to divide them we have to move to rational
A rational function in the variable is a function the form where and are
polynomial functions, and is not the constant zero function. The domain of a
rational function is all real numbers except for where the denominator is equal to
Which of the following are rational functions?
All polynomials can be thought of as rational functions.
Working with rational functions
Find the domain of the rational function
We start by setting the denominator equal
The domain of is all except these two values. Thus:
When we need to simplify the form of a rational expression, our approach depends on
the particular form we are presented with. If it consists of only a single fraction, we
divide out the common factors.
Simplify the expression .
We begin by factoring.
The numerator factors as , and the denominator factors as . That means,
When dealing with a sum/difference of two fractions, we must first convert to a
common denominator. After the addition, we can divide away any common factors
that are still present.
Simplify the expression
We’ll start by factoring the
denominators. , and . These two fractions have a common denominator of .
Once we convert each fraction to the common denominator, we added numerators,
If we have a complex fraction, involving a fraction in the numerator or the
denominator, we can start by multiplying the numerator and denominator of the big
fraction, by the common denominator of the smaller fractions. That eliminates the
smaller fractions leaving the outer one to deal with.
Simplify the expression
The smaller fractions in the numerator and denominator have a common
denominator . We begin my multiplying both the numerator and denominator by .
There are no common factors between the numerator and denominator, so this
cannot be simplified any further.
For the rational function , find and simplify the following:
means replace in the formula for with . This gives:
The quantity is frequently referred to as the Difference Quotient. We’ll be seeing
much more of the difference quotient soon.