Problem 108 - Algebra and Trigonometry

Question

What is a rational function?

Step-by-Step Solution

Answer

A rational function is an function represented as the quotient of two polynomials, in the form \(f(x) = \frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials and \(Q(x) ≠ 0\). The denominator cannot be zero as this would make the function undefined.

Step 1: Define a polynomial

A polynomial can be defined as an expression consisting of variables and coefficients, involving terms in the form \(a_nx^n\), where \(a_n\) is the coefficient and \(n\) is the power. The sum of these 'n' terms can be zero, a natural number or a negative integer, but it cannot be a fraction.

Step 2: Rational function definition

A rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The denominator can't be zero because it would make the function undefined.

Step 3: General form of a rational function

The general form of a rational function is \(f(x) = \frac{P(x)}{Q(x)}\) where \(P(x)\) and \(Q(x)\) are polynomials and \(Q(x) ≠ 0\).