2-6-7 Rational Functions Explained
Key Concepts of Rational Functions
Rational functions are a specific type of function where both the numerator and the denominator are polynomials. Key concepts include:
- Definition of Rational Functions: Functions of the form \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials.
- Vertical Asymptotes: Lines where the function approaches infinity or negative infinity.
- Horizontal Asymptotes: Lines where the function approaches as \( x \) approaches infinity or negative infinity.
- Holes: Points where the function is undefined but can be simplified to remove the discontinuity.
- Domain and Range: The set of all possible inputs (domain) and outputs (range) of a rational function.
1. Definition of Rational Functions
A rational function is defined as the ratio of two polynomials. For example, \( f(x) = \frac{3x + 2}{x^2 - 4} \) is a rational function.
Example:
Identify the numerator and denominator in the rational function \( \frac{5x^2 - 3x + 1}{2x + 7} \):
The numerator is \( 5x^2 - 3x + 1 \) and the denominator is \( 2x + 7 \).
2. Vertical Asymptotes
Vertical asymptotes occur where the denominator of the rational function is zero, and the numerator is not zero. These are lines where the function approaches infinity or negative infinity.
Example:
Find the vertical asymptote of \( f(x) = \frac{1}{x - 2} \):
Set the denominator equal to zero: \( x - 2 = 0 \Rightarrow x = 2 \).
The vertical asymptote is \( x = 2 \).
3. Horizontal Asymptotes
Horizontal asymptotes describe the behavior of the function as \( x \) approaches infinity or negative infinity. They depend on the degrees of the polynomials in the numerator and the denominator.
Example:
Find the horizontal asymptote of \( f(x) = \frac{3x^2 + 2}{x^2 - 4} \):
Since the degrees of the numerator and the denominator are the same, the horizontal asymptote is \( y = \frac{3}{1} = 3 \).
4. Holes
Holes occur when the numerator and the denominator have common factors that can be canceled. These points are undefined but can be simplified to remove the discontinuity.
Example:
Find the hole in the function \( f(x) = \frac{x^2 - 4}{x - 2} \):
Factor the numerator: \( x^2 - 4 = (x - 2)(x + 2) \).
Cancel the common factor: \( \frac{(x - 2)(x + 2)}{x - 2} = x + 2 \).
The hole occurs at \( x = 2 \), and the simplified function is \( y = x + 2 \).
5. Domain and Range
The domain of a rational function is all real numbers except those that make the denominator zero. The range is the set of all possible outputs, excluding values that create vertical asymptotes or holes.
Example:
Find the domain and range of \( f(x) = \frac{1}{x - 3} \):
The domain is all real numbers except \( x = 3 \).
The range is all real numbers except \( y = 0 \).
Examples and Analogies
To better understand rational functions, consider the following analogy:
Imagine a rational function as a road with potholes (holes) and bridges (asymptotes). The road represents the function, and the potholes are points where the road is temporarily closed. The bridges are places where the road stretches infinitely high or low.
Practical Applications
Rational functions are used in various real-world applications, such as:
- Modeling population growth with limited resources.
- Analyzing economic trends with constraints.
- Solving engineering problems involving ratios and proportions.
Example:
In economics, the demand function \( D(p) = \frac{100}{p - 5} \) models the relationship between price \( p \) and demand. Understanding the vertical asymptote at \( p = 5 \) helps in predicting market behavior.