What is a Rational Function?

A rational function is a type of function that can be expressed as the ratio of two polynomials. In simpler terms, it’s a fraction where the numerator and the denominator are both polynomials. The only restriction is that the denominator cannot be zero, as division by zero is undefined.

General Form

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)
eq 0 $

Examples of Rational Functions

1. $f(x) = frac{2x + 1}{x – 3}$
2. $g(x) = frac{x^2 + 4}{x^2 – 1}$
3. $h(x) = frac{3x^3 – x}{2x^2 + 5}$

Key Characteristics

Domain

The domain of a rational function is all real numbers except the values that make the denominator zero. For instance, in $f(x) = frac{2x + 1}{x – 3}$, the denominator becomes zero when $x = 3$. Hence, the domain is all real numbers except $x = 3$

Vertical Asymptotes

Vertical asymptotes occur at the values of $x$ that make the denominator zero (and are not canceled out by the numerator). These are the points where the function tends to infinity. For example, $f(x) = frac{1}{x – 2}$ has a vertical asymptote at $x = 2$

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of the function as $x$ approaches infinity or negative infinity. To determine the horizontal asymptote, compare the degrees of the numerator and the denominator:

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $y = 0$
• If the degrees are equal, the horizontal asymptote is $y = frac{a}{b}$, where $a$ and $b$ are the leading coefficients of the numerator and denominator, respectively.
• If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

Holes

Holes occur in the graph of a rational function when a factor in the numerator and the denominator cancels out. For example, in $f(x) = frac{(x – 1)(x + 2)}{(x – 1)(x + 3)}$, there is a hole at $x = 1$

Conclusion

Understanding rational functions is essential in algebra and calculus. They appear in various real-world applications, such as modeling rates and proportions. By studying their properties, like domain, asymptotes, and holes, you can gain a comprehensive understanding of their behavior and how to graph them effectively.

Citations

1. 1. Khan Academy – Rational Functions
2. 2. Math is Fun – Rational Functions
3. 3. Paul’s Online Math Notes – Rational Functions