Rational functions are a powerful tool in algebra, combining polynomials in fractions. They're used to model real-world scenarios like work rates and chemical concentrations. Understanding their behavior is key to solving complex problems.
Graphing rational functions involves finding asymptotes, intercepts, and analyzing behavior near critical points. This visual representation helps in understanding function behavior and solving related problems. Mastering these concepts opens doors to advanced mathematical analysis.
Rational Function Notation and Domain
Arrow notation for rational functions
Rational functions written as f(x)=Q(x)P(x) where P(x) and Q(x) are polynomial functions
P(x) represents the numerator polynomial
Q(x) represents the denominator polynomial
expresses rational functions as x↦Q(x)P(x)
Maps input values x to output values Q(x)P(x)
Example: x↦x−32x+1 maps input x to output x−32x+1
Domain of rational functions
Domain includes all real numbers except those making the denominator equal to zero
Solve Q(x)=0 to find excluded values
Example: For f(x)=x+2x2−1, solve x+2=0 to get x=−2 as the excluded value
Domain is the set of all real numbers minus the excluded values
Example: For f(x)=x−13, the domain is R∖{1} (all real numbers except 1)
Asymptotes and Graphing Rational Functions
Asymptotes of rational functions
Vertical asymptotes occur when the denominator equals zero and the numerator does not
Found by solving Q(x)=0
Example: For f(x)=x2−4x+1, vertical asymptotes at x=−2 and x=2
Horizontal asymptotes depend on the degrees of the numerator and denominator polynomials
If degree of numerator < degree of denominator, is y=0
If degrees are equal, horizontal asymptote is y=bnan (an and bn are leading coefficients)
If degree of numerator = degree of denominator + 1, oblique (slant) asymptote exists
If degree of numerator > degree of denominator + 1, no horizontal or oblique asymptote
Graphing rational functions
Find the domain and plot points of discontinuity (vertical asymptotes)
Determine horizontal or oblique asymptotes
Find x- and y-intercepts by setting the numerator and denominator equal to zero
Analyze function behavior near asymptotes and intercepts
Plot additional points to complete the graph
Example: To graph f(x)=x−2x+1, find vertical asymptote at x=2, horizontal asymptote at y=1, x-intercept at (−1,0), and y-intercept at (0,−21)
Behavior near asymptotes
As x approaches a vertical asymptote, function values approach positive or negative infinity
Sign depends on signs of numerator and denominator near the asymptote
Example: For f(x)=x−11, as x→1−, f(x)→−∞, and as x→1+, f(x)→+∞
As x approaches ±∞, function values approach the horizontal asymptote (if it exists)
Function may approach asymptote from above or below based on signs of leading coefficients
Example: For f(x)=x−32x+1, as x→±∞, f(x)→2 (approaching from above)
of rational functions is determined by analyzing limits as x approaches infinity
Real-world applications of rational functions
Model inverse proportionality, concentration of solutions, and Boyle's Law (pressure-volume relationship)
Example: Rate of work problems (if 3 workers take 4 hours, how long for 6 workers?)
Example: Mixing solutions (creating a 60% alcohol solution from 90% and 30% solutions)
Steps to solve application problems:
Identify given information and unknown quantity
Set up a rational function to model the situation
Solve the equation or interpret the graph to answer the question
Example: A group can paint a house in 10 hours. How long will it take if 3 people leave the group, which originally had 12 people? Model with t=n120, where t is time in hours and n is the number of people. With 9 people, t=9120≈13.33 hours.
Advanced Concepts in Rational Functions
Continuity and Limits
Continuity of rational functions depends on the domain and behavior at potential discontinuities
Limits help analyze function behavior near asymptotes and potential discontinuities
Removable discontinuities occur when a factor cancels in the numerator and denominator
Function Composition
Rational functions can be composed with other functions to create more complex relationships
Analyze domain and range carefully when composing functions
Key Terms to Review (5)
Arrow notation: Arrow notation is a way to describe the behavior of a function as the input approaches a particular value or infinity. It is commonly used to express limits, such as $\lim_{{x \to c}} f(x)$.
End behavior: End behavior describes the behavior of the graph of a function as the input values approach positive or negative infinity. It helps to understand how the function behaves at the extremes of its domain.
Horizontal asymptote: A horizontal asymptote is a horizontal line that a graph of a function approaches as the input (x) either increases or decreases without bound. This line represents a value that the function will get infinitely close to but never actually reach.
Reciprocal function: A reciprocal function is a function of the form $f(x) = \frac{1}{g(x)}$, where $g(x)$ is a non-zero polynomial. The simplest example is $f(x) = \frac{1}{x}$.
Removable discontinuity: A removable discontinuity occurs in a function at a point where the function is not defined but can be made continuous by redefining the function at that point. It often appears as a 'hole' in the graph of the function.