Problem 108 - College Algebra

Question

What is a rational function?

Step-by-Step Solution

Answer

A rational function is a mathematical function that can be expressed as the quotient of two polynomial functions.

Step 1: Definition of a Function

In math, a function is a rule that relates an input to an output. It means for each input, you will get one output and this mapping of input to output should be clearly specified.

Step 2: Introduction to Polynomial Functions

A polynomial function is a function that can be expressed in the form \( f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_2x^2 + a_1x + a_0 \), where \( n \) is a nonnegative integer and the \( a_i \)'s are the coefficients.

Step 3: Definition of a Rational Function

A rational function is a function of the form \( f(x) = \frac{P(x)}{Q(x)} \) where \( P(x) \) and \( Q(x) \) are polynomial functions. Rational functions are versatile and can express a wide variety of mathematical situations. It's important to note that, in a rational function, the denominator \( Q(x) \) cannot be zero because division by zero is undefined.