rational functions their graphs

Rational Functions

Rational Function: A type of function that can be expressed as the ratio of two polynomial functions. This encapsulates the concept that, like rational numbers, rational functions can be represented as a division of two quantities, allowing for the examination of their behavior in various mathematical contexts.
General Form: A rational function can be expressed in the form( f(x) = \frac{P(x)}{Q(x)} ) where ( Q(x)
eq 0 ).

( f(x) = \frac{5}{x} ): A simple rational function where the numerator is a constant.
( g(x) = \frac{x^2 - 2x + 1}{x^2 + 7x - 4} ): A rational function with polynomial expressions in both the numerator and denominator, demonstrating the typical form of a rational function.
( h(x) = \frac{2x^2 + 7x - 4}{5x - 2} ): This function represents a more complex polynomial ratio, useful for further analysis in calculus or algebraic studies.

Domain Restrictions: The domain of a rational function is defined by the values of the variable for which the function is defined. Since division by zero is undefined, it is essential to determine where the denominator equals zero, as these values will be excluded from the domain.
To find values where the function is undefined, set the denominator equal to zero and solve for x.

For the function ( f(x) = \frac{3}{x - 2} ):
- Set denominator to zero: ( x - 2 = 0 \Rightarrow x = 2 ).
- Hence, the function is undefined at ( x = 2 ).

Factorization is crucial for simplifying rational functions and can reveal additional domain restrictions.
- Example: For ( f(x) = \frac{x^2 - 4}{x^2 - 1} ), one would factor both the numerator and the denominator:
  - ( x^2 - 4 = (x - 2)(x + 2) ) and ( x^2 - 1 = (x - 1)(x + 1) ).
  - Identify restrictions: the function is undefined for values ( x = -1, 1, 2, -2 ).

Definition: Vertical asymptotes represent the values of x where the function approaches infinity or negative infinity, indicating that the function does not touch or cross the line.
Occurrence: They occur at values that restrict the domain due to denominator zeroes.

Example Function: For ( f(x) = \frac{2}{x - 3} ):
- Set the denominator to zero: ( x - 3 = 0 \Rightarrow x = 3 ).
- Vertical asymptote occurs at ( x = 3 ).

Definition: Horizontal asymptotes are horizontal lines that the graph of the function approaches as ( x ) approaches infinity or negative infinity.
Conditions for Horizontal Asymptotes:
1. If the degree of ( P(x) < Q(x) ), then the horizontal asymptote is ( y = 0 ).
2. If the degree of ( P(x) = Q(x) ), then the horizontal asymptote is the ratio of the leading coefficients of P and Q.
3. If the degree of ( P(x) > Q(x) ), then the horizontal asymptote does not exist.

For the function ( f(x) = \frac{2x^2 + 5}{3x^2 + 4} ):
- Since the degrees of the numerator and denominator are equal, the horizontal asymptote is ( y = \frac{2}{3} ).

Definition: An oblique (or slant) asymptote occurs when the degree of the numerator is exactly one greater than the degree of the denominator. This type of asymptote is not horizontal but instead slants at an angle.
Finding the Equation: The equation of the oblique asymptote is determined through polynomial long division.

Example Function: For ( f(x) = \frac{2x^3 + x^2}{x^2 + 3} ):
- Perform polynomial long division to find the asymptote.
- Result: The asymptote can be represented in the form ( y = mx + b ) where m and b are specific values derived from the division.

Procedures for Sketching Graphs:
1. Identify vertical asymptotes and plot them.
2. Identify horizontal or oblique asymptotes and draw them.
3. Calculate x-intercepts by setting the numerator to 0.
4. Calculate y-intercepts by setting x = 0.
5. Plot additional points between intercepts and asymptotes to provide a complete picture of the function's behavior.

For the function ( f(x) = \frac{3x + 4}{x - 1} ):
- Calculate asymptotes: Vertical at ( x = 1 ), Horizontal at ( y = 3 ).
- Sketch the graph by plotting the intercepts and asymptotic behavior.

Example 1: Consider the average cost of a zoo membership. The average cost function could be modeled as ( C(x) = \frac{100}{x} + 1 ), where ( x ) represents the number of visits. If ( x = 100 ), then ( C = \frac{100}{100} + 1 = 2 ), indicating the average cost per visit decreases with more visits.
Example 2: For renting a car, an average cost function could be modeled as ( C(x) = \frac{19 + 0.30x}{x} ). To achieve an average cost of $0.35, one must solve for maximum mileage conditions, thus exposing the interplay between fixed costs and variable usage.
Conclusion: Graphical analysis, including focusing on limits and asymptotes, is pivotal in understanding cost dynamics that come into play with continued usage, providing valuable insights into rational functions.