Rational Function & Rational Number

Rational Function:

In past grades, we learnt the concept of rational number. It is the quotient or ratio of two integers, where the denominator is not equal to zero. Hence, the name rational is derived from the word ratio. Formally, it is defined as,

Definition 1: A number that can be expressed in the form of \( \frac pq \) where p and q are integers and q ≠ 0 , is a rational number.Being mathematicians, one of our likes is to extend a given logic and see all possible circumstances wherever it can be applied. In this case, the logic can be extended to understand what rational function is. It is the ratio of two polynomial functions with the denominator polynomial not equal to zero. Just like rational numbers, rational function definition is given as:

Definition 2: A rational function R(x) is the function in the form\( \frac{ P(x)}{Q(x)}\)  where P(x) and Q(x) are polynomial functions and Q(x) is a non-zero polynomial.

R(x) = \( \frac {P(x)}{Q(x)}\) ,Q(x) ≠ 0

From the given condition for Q(x),we can conclude that zeroes of the polynomial function in the denominator do not fall in the domain of the function. When Q(x) = 1, i.e. a constant polynomial function, the rational function becomes a polynomial function.

Graphing Rational Functions:

One very important concept for graphing rational functions is to know about their asymptotes. An asymptote is a line or curve which stupidly approaches the curve forever but yet never touches it. In fig. 1, an example of asymptotes is given.

Rational function

Figure 1: Asymptotes

Rational functions can have 3 types of asymptotes:

    • Horizontal Asymptotes: This literally means that the asymptote is horizontal i.e. parallel to the axis of the independent variable. R(x) can only have a horizontal asymptote if

      Degree of P(x) ≤ Degree of Q(x)

      To determine the asymptotes, divide the numerator and the denominator of R(x) by \( x^{Degree~of~Q(x)} \)  . After that, find the value R(x) approaches as x tends to a very large value. This value gives the height of the asymptote.

    • Vertical Asymptotes: R(x) will have vertical asymptotes at the zeros ofQ(x). This is because at the zeros of Q(x), Q(x)=0.This means that just towards left and right of the zero of Q(x), the value of Q will be very small negative and positive number respectively. This means that value of R(x) will be a large negative and positive number respectively towards just left and right of that point.
  • Oblique Asymptotes: R(x) will have oblique asymptote if it can be represented in the form \((x)~+~\frac{1}{Q(x)}\) . When Q(x) ≫ 0, R(x) ≈ T(x). The curve or line T(x) hence becomes an oblique asymptote.

To quote an example, let us take R(x) = \( \frac{x^2+3x+3}{x+1}\).

Here, degree of P(x) is greater than that of Q(x). So, it can’t have a horizontal asymptote. But it will have a vertical asymptote at x=-1. This is because that point is the zero of its denominator polynomial.

It can also be written as R(x) = \( (x+2)~+~\frac{1}{x+1}\) . So, when x ≫ 0, R(x) ≈ x + 2. So, y = x + 2 will be an oblique asymptote. The graph of the function and all the asymptotes are shown in fig. 2.

Rational function

Figure 2: A rational function with its asymptotes

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Practise This Question

Let a,b,cϵR. If f(x)=ax2+bx+c be such that a+b+c=3 and f(x+y)=f(x)+f(y)+xy, x, yϵR, then 10n=1 f(n) is equal to