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College Algebra
Unit 5 Overview: Polynomial and Rational Functions
5.1 Quadratic Functions
5.2 Power Functions and Polynomial Functions
5.3 Graphs of Polynomial Functions
5.4 Dividing Polynomials
5.5 Zeros of Polynomial Functions
5.6 Rational Functions
5.7 Inverses and Radical Functions
5.8 Modeling Using Variation

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5.6 Rational Functions

Last Updated on June 24, 2024

Rational functions are fractions of polynomials that model real-world situations. They have unique features like asymptotes, holes, and intercepts that shape their graphs. Understanding these elements helps us analyze their behavior and solve practical problems.

We'll explore how to write, graph, and apply rational functions. We'll also dive into advanced concepts like limits and continuity. These skills are crucial for tackling complex problems in math and science.

Rational Function Fundamentals

Arrow notation for rational functions

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  • Rational functions written as the ratio of two polynomial functions f(x)=P(x)Q(x)f(x) = \frac{P(x)}{Q(x)}, where P(x)P(x) and Q(x)Q(x) are polynomial functions and Q(x)≠0Q(x) \neq 0
  • Arrow notation expresses a function using an arrow x→P(x)Q(x)x \rightarrow \frac{P(x)}{Q(x)}
  • Example: f(x)=x+1x−2f(x) = \frac{x+1}{x-2} written as x→x+1x−2x \rightarrow \frac{x+1}{x-2}
  • Example: g(x)=3x2−4x+12x+5g(x) = \frac{3x^2-4x+1}{2x+5} written as x→3x2−4x+12x+5x \rightarrow \frac{3x^2-4x+1}{2x+5}

Domain of rational functions

  • Domain consists of all real numbers except those that make the denominator equal to zero
    • Find the domain by setting the denominator equal to zero and solving for xx
    • Values of xx that make the denominator zero are excluded from the domain
  • Example: For f(x)=x+1x−2f(x) = \frac{x+1}{x-2}, set x−2=0x-2=0 and solve for xx. Solution is x=2x=2, so the domain is R∖{2}\mathbb{R} \setminus \{2\}
  • Example: For g(x)=3x2−4x+12x+5g(x) = \frac{3x^2-4x+1}{2x+5}, set 2x+5=02x+5=0 and solve for xx. Solution is x=−52x=-\frac{5}{2}, so the domain is R∖{−52}\mathbb{R} \setminus \{-\frac{5}{2}\}

Graphing Rational Functions

Asymptotes of rational functions

  • Vertical asymptotes occur where the denominator of the rational function equals zero
    • xx-values that make the denominator zero are the locations of the vertical asymptotes
  • Horizontal asymptotes occur when the degree of the numerator is less than or equal to the degree of the denominator
    • Horizontal asymptote is y=0y=0 if the degree of the numerator is less than the degree of the denominator
    • If the degree of the numerator equals the degree of the denominator, the horizontal asymptote is y=anbny=\frac{a_n}{b_n}, where ana_n and bnb_n are the leading coefficients of the numerator and denominator, respectively
  • Example: For f(x)=2x+1x−3f(x) = \frac{2x+1}{x-3}, the vertical asymptote is at x=3x=3, and the horizontal asymptote is y=2y=2
  • Example: For g(x)=x2−4x+2g(x) = \frac{x^2-4}{x+2}, the vertical asymptote is at x=−2x=-2, and the horizontal asymptote is y=x−2y=x-2

Graphing rational functions

  • Steps to graph a rational function:
    1. Find the domain and locate any vertical asymptotes
    2. Locate any horizontal asymptotes
    3. Find the xx- and yy-intercepts, if any
      • To find xx-intercepts, set the numerator equal to zero and solve for xx
      • To find the yy-intercept, evaluate the function at x=0x=0
    4. Plot the asymptotes and intercepts, and sketch the curve accordingly
  • Example: To graph f(x)=x+1x−2f(x) = \frac{x+1}{x-2}:
    1. Domain is R∖{2}\mathbb{R} \setminus \{2\}, vertical asymptote at x=2x=2
    2. Horizontal asymptote at y=1y=1
    3. xx-intercept at (−1,0)(-1, 0), no yy-intercept
    4. Plot the asymptotes, intercept, and sketch the curve

Features of rational function graphs

  • Holes occur when a factor in the numerator and denominator cancels out
    • To find the location of a hole, factor the numerator and denominator and cancel common factors
    • The xx-value of the canceled factor is the location of the hole
  • End behavior describes the behavior of the graph as xx approaches positive or negative infinity
    • If the degree of the numerator is less than the degree of the denominator, the end behavior will approach the horizontal asymptote
    • If the degree of the numerator equals the degree of the denominator, the end behavior will approach the horizontal asymptote
    • If the degree of the numerator is greater than the degree of the denominator, the end behavior will approach positive or negative infinity, depending on the signs of the leading coefficients
  • Example: For f(x)=(x−1)(x+2)(x−1)(x−3)f(x) = \frac{(x-1)(x+2)}{(x-1)(x-3)}, there is a hole at x=1x=1
  • Example: For g(x)=x2+1x−2g(x) = \frac{x^2+1}{x-2}, the end behavior approaches positive infinity as x→∞x \rightarrow \infty and negative infinity as x→−∞x \rightarrow -\infty

Applications of Rational Functions

Applications of rational functions

  • Rational functions model various real-world situations
    • Rate problems (work rate, flow rate)
    • Mixture problems
    • Density problems
  • Steps to solve real-world problems using rational functions:
    1. Identify the given information and the unknown quantity
    2. Set up a rational function that models the situation
    3. Solve the rational function for the unknown quantity
    4. Interpret the result in the context of the problem
  • Example: If a pipe can fill a tank in 4 hours and another pipe can fill the same tank in 6 hours, how long will it take to fill the tank if both pipes are used simultaneously?
    1. Given: Pipe 1 fills the tank in 4 hours, Pipe 2 fills the tank in 6 hours, unknown is the time to fill the tank using both pipes
    2. Rational function: 14+16=1t\frac{1}{4} + \frac{1}{6} = \frac{1}{t}, where tt is the unknown time
    3. Solve: 14+16=1t→512=1t→t=125=2.4\frac{1}{4} + \frac{1}{6} = \frac{1}{t} \rightarrow \frac{5}{12} = \frac{1}{t} \rightarrow t = \frac{12}{5} = 2.4 hours
    4. Interpret: It will take 2.4 hours to fill the tank using both pipes simultaneously

Advanced Concepts in Rational Functions

Limits, Continuity, and Discontinuity

  • Limits describe the behavior of a rational function as x approaches a specific value (including infinity)
  • Continuity occurs when a function has no breaks, jumps, or holes in its graph
  • Discontinuity in rational functions can be:
    • Removable (hole): occurs when a factor cancels out (as in function composition)
    • Jump: occurs at vertical asymptotes
  • Example: For f(x)=x2−1x−1f(x) = \frac{x^2-1}{x-1}, there is a removable discontinuity at x=1x=1

Key Terms to Review (48)

Arrow notation: Arrow notation is a way to describe the behavior of functions as the input approaches a particular value or infinity. It is often used to express limits and asymptotic behavior.
Associative property of multiplication: The associative property of multiplication states that the way in which numbers are grouped in a multiplication problem does not change the product. Mathematically, for any real numbers $a$, $b$, and $c$, $(a \cdot b) \cdot c = a \cdot (b \cdot c)$.
Domain: The domain of a function is the complete set of possible input values (x-values) that allow the function to work within its constraints. It specifies the range of x-values for which the function is defined.
Division Algorithm: The Division Algorithm for polynomials states that given any two polynomials, a dividend and a non-zero divisor, there exist unique quotient and remainder polynomials. The degree of the remainder polynomial is less than the degree of the divisor.
Inverse of a rational function: The inverse of a rational function is a function that reverses the effect of the original rational function. It essentially swaps the dependent and independent variables, solving for the input in terms of the output.
Horizontal asymptote: A horizontal asymptote is a horizontal line that a graph approaches as the input values go to positive or negative infinity. It indicates the end behavior of a function's output.
Leading coefficient: The leading coefficient of a polynomial is the coefficient of the term with the highest degree. It plays a crucial role in determining the end behavior of the polynomial function.
Long division: Long division is a method used to divide polynomials by another polynomial of lesser or equal degree. It involves repeated division, multiplication, and subtraction to obtain the quotient and remainder.
Partial fraction decomposition: Partial fraction decomposition is a method used to express a rational function as the sum of simpler fractions. This technique is particularly useful for integrating rational functions.
Lower limit of summation: The lower limit of summation is the starting index value in a summation notation, often denoted by $i=1$ or another integer. It indicates where the series begins.
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 $f(x) = \frac{1}{x}$.
Rational function: A rational function is a function that can be expressed as the quotient of two polynomials, where the denominator is not zero. It has the form $f(x) = \frac{P(x)}{Q(x)}$ where $P(x)$ and $Q(x)$ are polynomials.
Removable discontinuity: A removable discontinuity occurs at a point in a function where the limit exists, but the function is either not defined or defined differently. This type of discontinuity can be 'removed' by appropriately defining or redefining the function at that point.
Polynomial: A polynomial is an algebraic expression consisting of variables, coefficients, and non-negative integer exponents. It can be written in the form $a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$ where $a_n, a_{n-1}, ..., a_1, a_0$ are constants and $n$ is a non-negative integer.
X-intercept: The x-intercept is the point where a graph crosses the x-axis, where the y-coordinate is zero. It represents the solution(s) to an equation when $y = 0$.
Zeros: Zeros of a polynomial function are the values of the variable that make the function equal to zero. In other words, they are the solutions to the equation $P(x) = 0$.
Vertical asymptote: A vertical asymptote is a line $x = a$ where a rational function $f(x)$ approaches positive or negative infinity as $x$ approaches $a$. It represents values that $x$ cannot take, causing the function to become unbounded.
Asymptote: An asymptote is a line or curve that a graph approaches but never touches. It represents the limit of a function's behavior as the input variable approaches a particular value. Asymptotes are an important concept in various mathematical topics, including rational expressions, functions, rational functions, exponential functions, logarithmic functions, and exponential and logarithmic models.
Arrow Notation: Arrow notation is a graphical representation used in the context of rational functions to depict the behavior and properties of a function. It provides a visual aid to understand the end behavior, asymptotes, and other key characteristics of a rational function.
Continuity: Continuity is a fundamental concept in mathematics that describes the smooth and uninterrupted behavior of a function or graph. It is a crucial property that ensures a function's values change gradually without any abrupt jumps or breaks.
Denominator: The denominator is the bottom part of a fraction that indicates the number of equal parts into which the whole has been divided. It represents the total number of parts being considered in a rational expression.
Discontinuity: Discontinuity refers to a break or interruption in the continuity of a function. It occurs when a function is not defined at a particular point or when the function exhibits a sudden jump or change in its value at a specific point within its domain.
End Behavior: The end behavior of a function refers to how the function behaves as the input variable approaches positive or negative infinity. It describes the limiting values or patterns that the function exhibits as it extends towards the far left and right sides of its graph.
Domain: The domain of a function refers to the set of input values for which the function is defined. It represents the range of values that the independent variable can take on, and it is the set of all possible values that can be plugged into the function to produce a meaningful output.
Division: Division is the mathematical operation of splitting a quantity into equal parts or groups. It is the inverse of multiplication and is used to determine how many times one number is contained within another.
Factor: A factor is a number or expression that divides another number or expression without a remainder. Factors are fundamental building blocks that can be used to break down and understand more complex mathematical expressions, particularly in the context of rational functions.
Hole: In the context of rational functions, a hole refers to a point on the graph where the function is undefined or discontinuous. This occurs when the numerator and denominator of the rational function share a common factor, resulting in a point where the function cannot be evaluated.
Function Composition: Function composition is the process of combining two or more functions to create a new function. The output of one function becomes the input of the next function, allowing for the creation of more complex mathematical relationships.
Horizontal Asymptote: A horizontal asymptote is a horizontal line that a function's graph approaches as the input value (x) approaches positive or negative infinity. It represents the limit of the function as it gets closer and closer to this line, without ever touching it.
Leading Coefficient: The leading coefficient of a polynomial is the numerical coefficient of the term with the highest degree. It is the first, or leading, coefficient in the standard form of a polynomial expression. The leading coefficient plays a crucial role in understanding and analyzing various topics in college algebra, including polynomials, quadratic equations, functions, and more.
Limit: A limit is a value that a function or sequence approaches as the input variable approaches a certain value. It represents the behavior of a function or sequence as it gets closer and closer to a specific point, without necessarily reaching that point.
Linear-Over-Linear: Linear-over-linear is a type of rational function where the numerator and denominator are both linear functions. This means the numerator and denominator are both first-degree polynomials, with variables raised to the power of 1. The structure of a linear-over-linear rational function allows for unique properties and behaviors that are important to understand in the context of rational functions.
Multiplication: Multiplication is a mathematical operation that involves the repeated addition of a number to itself. It is a fundamental concept that is essential in various areas of mathematics, including algebra, rational expressions, rational functions, and the polar form of complex numbers.
Long Division: Long division is a step-by-step procedure for dividing one polynomial by another, where the divisor is of a higher degree than the dividend. It involves repeatedly subtracting multiples of the divisor from the dividend until the remainder is of a lower degree than the divisor.
Numerator: The numerator is the top number in a fraction, which represents the quantity or number of parts being considered. It is an essential component of rational functions, as it helps determine the behavior and characteristics of the function.
Partial Fraction Decomposition: Partial fraction decomposition is a technique used in calculus and algebra to express a rational function as a sum of simpler rational functions. It is a fundamental tool in the study of integration and the analysis of rational expressions.
Polynomial: A polynomial is an algebraic expression that consists of variables and coefficients, where the variables are raised to non-negative integer powers. Polynomials are fundamental in various areas of mathematics, including algebra, calculus, and the study of functions.
Quadratic-over-Linear: A quadratic-over-linear function is a rational function where the numerator is a quadratic expression and the denominator is a linear expression. This type of function is important in the study of rational functions as it exhibits unique characteristics and behaviors that are crucial to understand.
Rational Function: A rational function is a function that can be expressed as the ratio of two polynomial functions. It is a function that can be written in the form $f(x) = P(x)/Q(x)$, where $P(x)$ and $Q(x)$ are polynomial functions and $Q(x)$ is not equal to zero.
Reciprocal: The reciprocal of a number is the value obtained by dividing 1 by that number. It represents the inverse relationship between two quantities, where one quantity is the multiplicative inverse of the other.
Remainder Theorem: The Remainder Theorem is a fundamental principle in polynomial division that states the relationship between the division of a polynomial by a linear expression and the value of the polynomial when the variable is set to a specific value. It provides a way to determine the remainder of a polynomial division without actually performing the long division process.
Slant Asymptote: A slant asymptote is a line that a rational function's graph approaches as the independent variable approaches positive or negative infinity. It represents the oblique line that the graph of the rational function tends to parallel as it moves further from the origin.
Simplification: Simplification is the process of reducing the complexity of an expression or equation by applying various mathematical rules and techniques to obtain a simpler, more manageable form. This concept is crucial in the context of rational expressions, rational functions, trigonometric identities, and trigonometric expressions, as it allows for more efficient calculations and better understanding of the underlying mathematical relationships.
Y-intercept: The y-intercept is the point at which a line or curve intersects the y-axis, representing the value of the dependent variable (y) when the independent variable (x) is zero. It is a crucial concept in understanding the behavior and properties of various mathematical functions and their graphical representations.
Vertical Asymptote: A vertical asymptote is a vertical line that a graph of a function approaches but never touches. It represents the vertical limit of the function's behavior, indicating where the function's value becomes arbitrarily large or small.
X-Intercept: The x-intercept of a graph is the point where the graph of a function or equation intersects the x-axis, indicating the value of x when the function's output or the equation's value is zero. The x-intercept is a crucial concept in understanding the behavior and properties of various mathematical functions and equations.
Zeros: Zeros, also known as roots, are the values of the independent variable that make a function equal to zero. They are the points where the graph of a function intersects the x-axis, representing the solutions to the equation f(x) = 0.
Range: In mathematics, the range refers to the set of all possible output values (dependent variable values) that a function can produce based on its input values (independent variable values). Understanding the range helps in analyzing how a function behaves and what values it can take, connecting it to various concepts like transformations, compositions, and types of functions.
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5.7 Inverses and Radical Functions