Q.Identify the following expressions as Rational Functions. Further classify them as Proper or Improper. If Improper, express them as sum of a polynomial and proper rational function.

Rational Function Classification – From Intuition to Precision

A rational function is simply a fraction where the numerator and denominator are both polynomials. Think of it like a ratio of two polynomials — hence the name rational (ratio). You already know fractions like $\frac{3}{4}$; now the top and bottom are algebraic expressions instead of plain numbers.

For example,

is a rational function. The numerator $2x+1$ is a polynomial of degree 1, and the denominator $x^2-4$ is a polynomial of degree 2.

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Why classify them?

The behaviour of a rational function — where it blows up, where it levels off, how it looks on a graph — depends critically on the degrees of the numerator and denominator. Classification tells you, at a glance, what kind of asymptotes the function has and how it behaves for very large $x$.

There are exactly three cases, based on comparing the degree of the numerator ($n$) and the degree of the denominator ($m$).

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The three types

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1. Proper rational function ($n < m$)

The denominator grows faster than the numerator. As $x \to \pm\infty$, the function approaches $0$. The $x$-axis ($y=0$) is a horizontal asymptote.

Example:

Here $n=1$, $m=2$, so $n<m$. As $x$ gets huge, the denominator dominates and $f(x) \to 0$.

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2. Improper rational function ($n \ge m$)

The numerator grows at least as fast as the denominator. The function does not tend to zero at infinity.

Case A: $n = m$

The leading terms dominate. The function approaches the ratio of the leading coefficients.

Example:

Both degrees are 2. Leading coefficients: $3$ and $1$. So as $x \to \pm\infty$, $f(x) \to \frac{3}{1} = 3$. The horizontal asymptote is $y=3$.

Case B: $n > m$

No horizontal asymptote exists. The function grows without bound (or goes to $-\infty$) as $x \to \pm\infty$.

If $n = m+1$, there is a slant (oblique) asymptote — a line that the function approaches. You find it by polynomial long division.

Example:

Here $n=2$, $m=1$, so $n = m+1$. Divide: $x^2+1$ divided by $x-2$ gives $x+2 + \frac{5}{x-2}$. The slant asymptote is $y = x+2$.

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Quick reference table

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The big picture

Classification is not just a naming exercise. It tells you instantly:

• Whether the function decays to zero (proper) or not (improper).
• Whether there is a horizontal line it approaches (proper or equal degrees).
• Whether you need to do polynomial division to find a slant asymptote.

When you later study integration, you'll see that proper rational functions are the ones you can decompose into partial fractions — another reason the classification matters deeply.