Understanding Rational Functions | Definition, Characteristics, and Simplification

rational function

A rational function is a mathematical function that can be expressed as the quotient of two polynomials

A rational function is a mathematical function that can be expressed as the quotient of two polynomials. In other words, it is a fraction of polynomial functions. The general form of a rational function is:

f(x) = P(x) / Q(x)

where P(x) and Q(x) are polynomials and Q(x) ≠ 0.

The domain of a rational function is all real numbers except for the values of x that make the denominator, Q(x), equal to zero. These values are called the vertical asymptotes, and they represent points where the function is not defined.

Rational functions can exhibit various behaviors based on the degrees and factors of their polynomials. Some common characteristics of rational functions include horizontal asymptotes, vertical asymptotes, holes (or removable discontinuities), and x-intercepts.

To simplify a rational function, you can factor the numerator and denominator and cancel out common factors. This can help identify any holes or asymptotes.

For example, consider the rational function:
f(x) = (2x^2 + 3x – 5) / (x^2 – 4)

Here, the numerator is a polynomial of degree 2 (quadratic), and the denominator is also a polynomial of degree 2. To find the vertical asymptotes, we set the denominator equal to zero and solve for x:

x^2 – 4 = 0
(x – 2)(x + 2) = 0
x = 2 or x = -2

Therefore, the vertical asymptotes for this function are x = 2 and x = -2.

To simplify the function, we can try factoring the numerator and denominator:

f(x) = [(2x – 1)(x + 5)] / [(x – 2)(x + 2)]

This reveals a hole at x = 1/2, where the factor (2x – 1) in the numerator cancels out with the factor (x – 2) in the denominator.

In conclusion, a rational function is a fraction of two polynomials, and it can have vertical asymptotes, horizontal asymptotes, holes, and x-intercepts. Simplifying a rational function involves factoring and canceling common factors.

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