1. What a Rational Function Means
A rational function is a function that can be written as a ratio of two polynomial functions. In other words, it looks like a fraction where both the numerator and the denominator are polynomials.
2. General Form of a Rational Function
Any rational function can be written as:
2.1. Expression
f(x) = \frac{P(x)}{Q(x)}
\text{where } P(x) \text{ and } Q(x) \text{ are polynomials and } Q(x) \ne 0
3. Domain of a Rational Function
A rational function is defined for all real values except where the denominator becomes zero. These points must be excluded from the domain.
3.1. Example
For
f(x) = \frac{1}{x - 3}
The denominator becomes zero at x = 3, so x = 3 is not allowed.
4. Examples of Rational Functions
Here are some typical rational functions:
4.1. Example 1
\( f(x) = \frac{2x + 3}{x - 1} \)
4.2. Example 2
\( f(x) = \frac{x^2 - 4}{x + 2} \)
Note: numerator factors to (x - 2)(x + 2); the graph has a removable hole at x = -2.
4.3. Example 3
\( f(x) = \frac{5}{x^2 + 1} \)
Denominator never becomes zero, so the domain is all real numbers.
4.4. Example 4
\( f(x) = \frac{x^3}{2x^2 - 8} \)
5. Properties of Rational Functions
Rational functions have certain predictable features that come from their algebraic form.
5.1. 1. Vertical Asymptotes
These occur where the denominator becomes zero and is not canceled by the numerator.
5.2. 2. Horizontal or Oblique Asymptotes
The long-term behaviour depends on the degrees of the numerator and denominator.
5.3. 3. Possible Holes
If a factor cancels between numerator and denominator, the graph has a hole at that x-value.
6. Why Rational Functions Matter
Rational functions appear in real-world models, physics formulas, and rate-related problems. Their asymptotic and discontinuous behaviour makes them useful for describing many natural processes.