
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 $4$ is not a factor of $7$, $\dfrac {7}{4}$ is not an integer. If we want to be able to divide integers, we have to move to the rational numbers, which are fractions $\dfrac {p}{q}$ where $p$ and $q$ are integers, and $q \ne 0$.

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 functions.

Which of the following are rational functions?
$f(x) = 0$ $f(x) = \frac {3x+1}{x^2-4x+5}$ $f(x)=e^x$ $f(x)=\frac {\sin (x)}{\cos (x)}$ $f(x) = -4x^{-3}+5x^{-1}+7-18x^2$ $f(x) = x^{1/2}-x +8$ $f(x)=\frac {\sqrt {x}}{x^3-x}$

### Working with rational functions

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.

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.

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.

The quantity $\frac {f(x+h)-f(x)}{h}$ is frequently referred to as the Difference Quotient. We’ll be seeing much more of the difference quotient soon.