MTH120 College Algebra Chapter 5.6

 5.6 Rational Functions

Rational functions are mathematical functions that can be expressed as the quotient or ratio of two polynomials. They are written in the form $�(�)=\frac{�(�)}{�(�)}$, where $�(�)$ and $�(�)$ are polynomials, and $�(�)$ is not the zero polynomial. In this section, we'll explore rational functions and their properties.

Key Concepts and Properties of Rational Functions:

1. Domain: The domain of a rational function is all real numbers except the values of $�$ that make the denominator $�(�)$ equal to zero. These values are called "excluded values" or "singularities."

2. Vertical Asymptotes: Vertical asymptotes are vertical lines $�=�$ where the rational function approaches infinity or negative infinity as $�$ approaches $�$ from one side. Vertical asymptotes occur at the excluded values of $�$ where $�(�)=0$.

3. Horizontal Asymptotes: Horizontal asymptotes are horizontal lines $�=�$ that the rational function approaches as $�$ goes to positive or negative infinity. The existence and location of horizontal asymptotes depend on the degrees of the polynomials $�(�)$ and $�(�)$.

4. Oblique (Slant) Asymptotes: Some rational functions have oblique asymptotes (slant asymptotes) when the degree of $�(�)$ is one more than the degree of $�(�)$. Oblique asymptotes are slant lines that the rational function approaches as $�$ goes to positive or negative infinity.

5. Vertical Holes: In some cases, a rational function may have holes in its graph, which occur when both the numerator and denominator have a common factor that cancels out at a certain point.

6. End Behavior: Understanding the end behavior of rational functions can help determine the behavior of the function as $�$ approaches infinity or negative infinity.

7. Graphing Techniques: Graphing rational functions involves identifying the vertical asymptotes, horizontal asymptotes, holes, and any other features that affect the shape of the graph.

8. Solving Rational Equations: Rational functions can be used to model real-world situations, and solving rational equations often involves finding the values of $�$ that make the function equal to a specific value.

Rational functions are commonly encountered in various fields of mathematics, science, engineering, and economics. They are used to model and analyze situations where one quantity depends on another in a non-linear or fractional way.

Understanding the properties and behaviors of rational functions is essential for solving problems involving these functions, graphing them accurately, and interpreting their real-world applications.

Arrow notation, also known as arrow diagram notation or interval notation, is a mathematical notation used to represent the domain and range of a function, intervals on the real number line, and inequalities. It uses arrows and brackets to describe sets of values in a concise and standardized way. Here's how arrow notation works:

1. Open Interval Notation:

• $�\in (�,�)$ means that $�$ is an element of the open interval between $�$ and $�$, excluding $�$ and $�$.

Example: $�\in (2,5)$ represents the set of all real numbers $�$ such that $2<�<5$.

2. Closed Interval Notation:

• $�\in [�,�]$ means that $�$ is an element of the closed interval between $�$ and $�$, including both $�$ and $�$.

Example: $�\in [2,5]$ represents the set of all real numbers $�$ such that $2\le �\le 5$.

3. Half-Open or Half-Closed Interval Notation:

• $�\in [�,�)$ means that $�$ is an element of the half-open interval that includes $�$ but not $�$.
• $�\in (�,�]$ means that $�$ is an element of the half-open interval that includes $�$ but not $�$.

Example: $�\in [2,5)$ represents the set of all real numbers $�$ such that $2\le �<5$.

4. Infinite Intervals:

• $�\in (-\mathrm{\infty },�)$ represents the set of all real numbers $�$ less than $�$.
• $�\in (�,\mathrm{\infty })$ represents the set of all real numbers $�$ greater than $�$.

Example: $�\in (-\mathrm{\infty },3)$ represents the set of all real numbers $�$ less than 3.

5. Combining Intervals:

You can use union ($\cup$) and intersection ($\cap$) symbols to combine intervals. For example:

• $�\in (-\mathrm{\infty },2)\cup (4,\mathrm{\infty })$ represents the set of all real numbers $�$ that are less than 2 or greater than 4.
• $�\in [1,3]\cap (2,4)$ represents the set of all real numbers $�$ that are in both the closed interval [1, 3] and the open interval (2, 4).

6. Inequalities:

Arrow notation is also used to represent inequalities:

• $�>�$ represents $�$ is greater than $�$.
• $�<�$ represents $�$ is less than $�$.
• $�\ge �$ represents $�$ is greater than or equal to $�$.
• $�\le �$ represents $�$ is less than or equal to $�$.

Example: $�>3$ represents the inequality $�>3$.

Arrow notation is particularly useful when describing sets of real numbers, intervals, and inequalities in a concise and standardized way, making it easier to communicate and work with mathematical concepts.

A vertical asymptote is a vertical line on a graph that a function approaches but never intersects as the independent variable (usually denoted as $�$) approaches certain values. Vertical asymptotes are commonly encountered in the graphs of rational functions and are associated with values of $�$ for which the denominator of the rational function becomes zero, causing the function to approach infinity or negative infinity.

Here are the key characteristics of vertical asymptotes:

1. Definition: A vertical asymptote occurs at $�=�$ if the function approaches positive or negative infinity as $�$ gets arbitrarily close to $�$, but it never reaches a specific value at $�=�$.

2. Rational Functions: Vertical asymptotes are frequently found in the graphs of rational functions of the form $�(�)=\frac{�(�)}{�(�)}$, where $�(�)$ and $�(�)$ are polynomials. Vertical asymptotes occur at values of $�$ for which $�(�)=0$ and $�(�)$ does not equal zero.

3. Excluded Values: The values of $�$ for which the denominator $�(�)$ becomes zero are called "excluded values" or "singularities." These excluded values are where vertical asymptotes may be present.

4. Behavior: As $�$ approaches an excluded value associated with a vertical asymptote, the function $�(�)$ may increase or decrease without bound, depending on whether the limit as $�$ approaches the excluded value from the left or right is positive or negative.

5. Vertical Line: A vertical asymptote is represented as a vertical line on the graph of the function at the $�$-coordinate corresponding to the excluded value.

6. Graph: Vertical asymptotes are typically depicted as dashed vertical lines on a graph to indicate that the function approaches but never crosses these lines.

Example:

Consider the rational function $�(�)=\frac{1}{�-2}$. In this case, there is a vertical asymptote at $�=2$ because the denominator $�-2$ becomes zero when $�=2$. As $�$ approaches 2 from the left ($�<2$), the function approaches negative infinity, and as $�$ approaches 2 from the right ($�>2$), the function approaches positive infinity. Therefore, there is a vertical asymptote at $�=2$, and the graph approaches this line but never crosses it.

Vertical asymptotes are important features to identify when graphing rational functions as they provide insights into the behavior of the function as $�$ approaches certain values. They also help determine the domain of the function and any discontinuities in its graph.

A horizontal asymptote is a horizontal line on the graph of a function that the function approaches as the independent variable (usually denoted as $�$) becomes very large in either the positive or negative direction. Horizontal asymptotes provide information about the long-term behavior of a function as $�$ approaches infinity or negative infinity.

Here are the key characteristics of horizontal asymptotes:

1. Definition: A horizontal asymptote occurs at $�=�$ if the function approaches the value $�$ as $�$ approaches infinity ($+\mathrm{\infty }$) or negative infinity ($-\mathrm{\infty }$). In other words, as $�$ becomes very large in magnitude, the function's values get closer and closer to $�$.

2. Behavior: If a function approaches a horizontal asymptote $�=�$ as $�$ goes to $\pm \mathrm{\infty }$, it means that the function levels off and stabilizes at or near the value $�$ as $�$ becomes very large.

3. Graph: A horizontal asymptote is represented as a horizontal line at the $�$-coordinate $�$ on the graph of the function. The function approaches this line as $�$ goes to $\pm \mathrm{\infty }$.

4. Types of Horizontal Asymptotes:

   • If $�$ is a finite number, the function approaches a horizontal asymptote at $�=�$. In this case, the function levels off and approaches a constant value as $�$ becomes very large.

   • If $�=+\mathrm{\infty }$ or $-\mathrm{\infty }$, the function approaches a horizontal asymptote at $�=+\mathrm{\infty }$ or $�=-\mathrm{\infty }$, respectively. In this situation, the function grows without bound as $�$ becomes very large.

   • If there is no horizontal asymptote, the function may exhibit other types of behavior as $�$ goes to $\pm \mathrm{\infty }$, such as oscillations or unbounded growth.

5. Rational Functions: Horizontal asymptotes are often encountered in the graphs of rational functions (quotients of polynomials). The degree of the numerator and denominator polynomials determines the behavior of the rational function and the existence of horizontal asymptotes.

6. Limits: To determine the horizontal asymptote of a function, you can evaluate the limit of the function as $�$ approaches $\pm \mathrm{\infty }$. If the limit exists and equals $�$, then $�=�$ is the horizontal asymptote.

Examples:

1. For the rational function $�(�)=\frac{3{�}^2-2�+1}{{�}^2+1}$, there is a horizontal asymptote at $�=3$ because, as $�$ approaches $\pm \mathrm{\infty }$, the ratio of the leading coefficients of the numerator and denominator is $3\mathrm{/}1$.

2. The function $�(�)={�}^{-�}$ has a horizontal asymptote at $�=0$ because the exponential term approaches zero as $�$ goes to $\mathrm{\infty }$.

Horizontal asymptotes are important for understanding the long-term behavior of functions and for determining the limits of functions as $�$ approaches $\pm \mathrm{\infty }$. They are valuable in applications where you need to analyze how a function behaves as its input becomes very large or very small.

Solving applied problems involving rational functions often requires the following steps:

1. Identify the Variables: Begin by identifying the variables involved in the problem and what they represent. Determine which quantity depends on another.

2. Set Up a Rational Function: Express the relationship between the variables using a rational function. This typically involves writing a function $�(�)$ where $�$ represents the independent variable (e.g., time) and $�(�)$ represents the dependent variable (e.g., distance, cost, population, etc.).

3. Identify the Numerator and Denominator: In the rational function, identify the numerator and denominator. These may represent different quantities or rates of change.

4. Analyze the Problem: Carefully read and understand the problem to identify any specific conditions, constraints, or requirements. Note any given values or initial conditions.

5. Solve for the Desired Quantity: Depending on the problem, you may need to solve for a specific quantity, such as finding the time at which two objects meet, the cost of a certain quantity of goods, or the population at a future time.

6. Set Up and Solve Equations: Translate the problem into mathematical equations using the rational function. This may involve setting the function equal to a given value or solving for a particular variable. If there are multiple variables, use the given information to set up equations that relate them.

7. Solve the Equations: Solve the equations algebraically. You may need to manipulate the equations, combine like terms, factor, or solve for a specific variable.

8. Consider Domain and Practical Implications: Check the domain of the rational function to ensure it makes sense in the context of the problem. Some values of $�$ may be excluded due to physical limitations or restrictions. Interpret the results in the context of the problem.

9. Check for Extraneous Solutions: In some cases, solving rational equations may yield extraneous solutions (solutions that do not satisfy the original problem). Verify that the solutions make sense in the context of the problem.

10. Present the Solution: Provide the solution to the problem in a clear and concise manner, including any relevant units of measurement.

Here are a few examples of problems involving rational functions:

Example 1: Distance-Time Problem A car is traveling at a constant speed of 60 miles per hour. How far will it travel in 3 hours?

Solution:

• Identify variables: Time ($�$), Distance ($�$)
• Set up the rational function: $�(�)=60�$
• Solve for $�(3)$: $�(3)=60\cdot 3=180$ miles
• Present the solution: The car will travel 180 miles in 3 hours.

Example 2: Cost Problem A company produces widgets at a cost of $5 per widget plus a fixed setup cost of $100. Express the total cost $�(�)$ of producing $�$ widgets as a rational function.

Solution:

• Identify variables: Number of widgets ($�$), Cost ($�(�)$)
• Set up the rational function: $�(�)=5�+100$
• Present the solution: $�(�)=5�+100$

Example 3: Population Growth A population of bacteria doubles in size every hour. If there are initially 100 bacteria, express the population $�(�)$ as a function of time $�$.

Solution:

• Identify variables: Time ($�$), Population ($�(�)$)
• Set up the rational function: $�(�)=100\cdot 2^{�}$
• Present the solution: $�(�)=100\cdot 2^{�}$

These are simplified examples, but real-world problems may involve more complex rational functions and require more advanced mathematical techniques. The key is to understand the problem, set up the appropriate rational function, and solve it in the context of the problem's requirements.

Finding the domain of a rational function involves determining the values of the independent variable ($�$) for which the function is defined and does not lead to division by zero. Here are the steps to find the domain of a rational function:

Step 1: Identify the Rational Function

Start by identifying the given rational function, typically expressed in the form $�(�)=\frac{�(�)}{�(�)}$, where $�(�)$ and $�(�)$ are polynomials, and $�(�)$ is not the zero polynomial.

Step 2: Identify Excluded Values

The excluded values for $�$ are the values that make the denominator ($�(�)$) equal to zero, as division by zero is undefined in mathematics. To find these excluded values, solve the equation $�(�)=0$ for $�$.

Step 3: Determine the Domain

The domain of the rational function consists of all real numbers except the excluded values you found in step 2. So, the domain $�$ can be expressed as:

$�=\{�\in \mathbb{�}\mid �\text{ does not equal the excluded values}\}$

Step 4: Consider Additional Restrictions (Optional)

In some cases, there may be additional restrictions on the domain based on the context of the problem or practical limitations. For example, a real-world problem involving distance and time may exclude negative values for time ($�$) because negative time doesn't make sense in that context.

Step 5: Present the Domain

Present the domain $�$ in interval notation or set notation, depending on the format required. Interval notation is often used when the domain is a contiguous interval on the real number line, while set notation is more general and can be used for any domain.

Here are a few examples to illustrate finding the domains of rational functions:

Example 1: Rational Function with No Excluded Values $�(�)=\frac{{�}^2-4}{{�}^2+3�+2}$

• Excluded values: Solve ${�}^2+3�+2=0$ to find that the excluded values are $�=-2$ and $�=-1$.
• Domain: $�=\{�\in \mathbb{�}\mid �\mathrm{\ne }-2,�\mathrm{\ne }-1\}$

Example 2: Rational Function with Additional Restrictions $�(�)=\frac{1}{\sqrt{�-3}}$

• Excluded values: The expression $\sqrt{�-3}$ is defined only for non-negative values under the square root ($�-3\ge 0$), so the excluded values are $�<3$.
• Domain: $�=\{�\in \mathbb{�}\mid �\ge 3\}$

Example 3: Rational Function with Infinite Excluded Values $ℎ(�)=\frac{1}{{�}^2-9}$

• Excluded values: Solve ${�}^2-9=0$ to find the excluded values $�=3$ and $�=-3$. However, there are no additional restrictions.
• Domain: $�=\{�\in \mathbb{�}\mid �\mathrm{\ne }3,�\mathrm{\ne }-3\}$

Finding the domain of a rational function is an important step in understanding the function's behavior and where it is defined. It ensures that you do not perform operations that lead to undefined results.

Vertical asymptotes are vertical lines on the graph of a rational function where the function approaches either positive or negative infinity as the independent variable ($�$) approaches certain values. These vertical lines are associated with values of $�$ for which the denominator of the rational function becomes zero, leading to vertical asymptotes. Here's how to identify vertical asymptotes in a rational function:

Step 1: Identify the Rational Function

Start by identifying the given rational function, typically expressed in the form $�(�)=\frac{�(�)}{�(�)}$, where $�(�)$ and $�(�)$ are polynomials, and $�(�)$ is not the zero polynomial.

Step 2: Find the Excluded Values

To find the vertical asymptotes, you need to find the values of $�$ for which the denominator ($�(�)$) becomes zero. Solve the equation $�(�)=0$ for $�$. These values are called "excluded values" or "singularities" because they exclude $�$ from the domain of the rational function.

Step 3: Identify Vertical Asymptotes

The vertical asymptotes occur at the values of $�$ that you found in step 2. These are the vertical lines on the graph where the function approaches either positive or negative infinity as $�$ approaches the excluded values. Each excluded value corresponds to a vertical asymptote.

Step 4: Determine the Behavior

Determine the behavior of the function as $�$ approaches each vertical asymptote. To do this, evaluate the limit of the function as $�$ approaches each excluded value from the left and right sides. If the limits differ, it indicates that the function approaches different infinities on either side of the vertical asymptote.

Step 5: Sketch the Graph

Once you've identified the vertical asymptotes and understood the behavior of the function near these asymptotes, you can sketch the graph of the rational function. Include the vertical asymptotes as dashed vertical lines on the graph.

Here are a few examples to illustrate how to identify vertical asymptotes in rational functions:

Example 1: Rational Function with Simple Vertical Asymptote $�(�)=\frac{1}{�-2}$

• Excluded value: Solve $�-2=0$ to find the excluded value $�=2$.
• Vertical asymptote: $�=2$ is a vertical asymptote.
• Behavior: As $�$ approaches 2 from the left, $�(�)$ approaches negative infinity, and as $�$ approaches 2 from the right, $�(�)$ approaches positive infinity.

Example 2: Rational Function with Multiple Vertical Asymptotes $�(�)=\frac{1}{{�}^2-4}$

• Excluded values: Solve ${�}^2-4=0$ to find the excluded values $�=2$ and $�=-2$.
• Vertical asymptotes: $�=2$ and $�=-2$ are vertical asymptotes.
• Behavior: As $�$ approaches 2 from the left, $�(�)$ approaches negative infinity, and as $�$ approaches 2 from the right, $�(�)$ approaches positive infinity. Similarly, as $�$ approaches -2 from the left, $�(�)$ approaches negative infinity, and from the right, it approaches positive infinity.

Identifying vertical asymptotes is essential for understanding the behavior of rational functions, particularly as $�$ approaches certain values. These asymptotes help determine the domain of the function and any discontinuities in its graph.

A removable discontinuity, also known as a removable singularity or a point of removable discontinuity, is a type of discontinuity that occurs at a specific point in the graph of a function but can be "removed" or "filled in" by redefining the function at that point. In other words, a removable discontinuity is a point where the function could have been defined differently to make it continuous.

Here are the characteristics and properties of removable discontinuities:

1. Definition: A removable discontinuity occurs at a point $(�,�(�))$ in the graph of a function if the limit of the function as it approaches $(�,�(�))$ exists, but the function's value at that point, $�(�)$, is not equal to the limit.

2. Mathematical Representation: Mathematically, a removable discontinuity can be represented as follows:

   ${\lim }_{�\to �}�(�)\text{ exists, but }�(�)\mathrm{\ne }{\lim }_{�\to �}�(�)$

3. Removability: The term "removable" implies that you can redefine the function at the point of discontinuity to make it continuous. This is typically done by assigning the function the value of the limit at that point. In other words, you "remove" the discontinuity by filling in the gap.

4. Graphical Representation: On a graph, a removable discontinuity appears as a "hole" or a "jump" in the function's curve at the point $(�,�(�))$. The curve approaches the hole from both sides, indicating that the limit exists, but the hole itself indicates the discontinuity.

5. Examples: Common examples of functions with removable discontinuities include rational functions in which the denominator becomes zero at a certain point. For instance, the function $�(�)=\frac{{�}^2-4}{�-2}$ has a removable discontinuity at $�=2$, which can be removed by redefining $�(2)$ to be the limit ${\lim }_{�\to 2}�(�)$.

6. Simplification: Identifying and removing removable discontinuities can simplify the analysis of a function and make it easier to find limits, calculate derivatives, or evaluate integrals.

7. Notation: Sometimes, removable discontinuities are indicated on a graph using an open circle at the point of discontinuity.

It's important to note that not all discontinuities are removable. Other types of discontinuities include jump discontinuities, infinite discontinuities, and essential discontinuities, which cannot be removed by redefining the function at the point of discontinuity. Removable discontinuities are a specific and relatively mild form of discontinuity that can be "repaired" to make the function continuous at that point.

Removable discontinuities are a common feature of rational functions. They occur when the denominator of a rational function becomes zero at a specific point, causing a hole or gap in the graph that can be "filled in" to make the function continuous. Here's how to identify and understand removable discontinuities in rational functions:

1. Identifying Removable Discontinuities:

• A removable discontinuity in a rational function $�(�)=\frac{�(�)}{�(�)}$ occurs when both the numerator $�(�)$ and the denominator $�(�)$ have a common factor that cancels out at a specific point, resulting in a hole in the graph.
• Mathematically, a removable discontinuity at $�=�$ can be identified when the denominator $�(�)=0$ and the numerator $�(�)$ is nonzero, leading to a canceled-out factor.

2. Mathematical Representation:

• A removable discontinuity can be represented as follows: ${\lim }_{�\to �}�(�)\text{ exists, but }�(�)\mathrm{\ne }{\lim }_{�\to �}�(�)$

3. Example:

• Consider the rational function $�(�)=\frac{{�}^2-4}{�-2}$. At $�=2$, the denominator $�-2$ becomes zero, leading to a removable discontinuity. To remove the discontinuity, you can simplify the function by canceling out the common factor $(�-2)$ in both the numerator and the denominator: $�(�)=\frac{(�-2)(�+2)}{�-2}$ $�(�)=�+2$
• The simplified function $�(�)=�+2$ does not have a discontinuity at $�=2$, as the hole has been "filled in."

4. Graphical Representation:

• On a graph, a removable discontinuity appears as a hole in the curve of the rational function. The curve approaches the hole from both sides, indicating that the limit exists, but the hole itself indicates the discontinuity.

5. Removability:

• The term "removable" implies that you can redefine the function at the point of discontinuity to make it continuous. In most cases, you fill in the gap by assigning the function the value of the limit at that point.

6. Algebraic Manipulation:

• To remove the discontinuity, you can often perform algebraic manipulation of the rational function to cancel out the common factors in the numerator and denominator. This simplifies the function, making it continuous at the point of discontinuity.

In summary, removable discontinuities in rational functions occur when there is a common factor that cancels out in the numerator and denominator, leading to a hole in the graph. These discontinuities can be "removed" by simplifying the function and filling in the gap to make it continuous. Identifying removable discontinuities is an essential step in analyzing rational functions and understanding their behavior.

Identifying horizontal asymptotes of rational functions is important for understanding the long-term behavior of the function as the independent variable ($�$) approaches positive or negative infinity. Horizontal asymptotes provide information about where the function levels off as $�$ becomes very large or very small. Here's how to identify horizontal asymptotes in rational functions:

Step 1: Identify the Rational Function

Start by identifying the given rational function, typically expressed in the form $�(�)=\frac{�(�)}{�(�)}$, where $�(�)$ and $�(�)$ are polynomials, and $�(�)$ is not the zero polynomial.

Step 2: Examine the Degrees of Numerator and Denominator

Horizontal asymptotes are determined by the degrees of the polynomials in the numerator ($�(�)$) and denominator ($�(�)$) of the rational function:

1. Degree of Numerator $�(�)$: Denoted as $�$, it is the highest power of $�$ in the numerator. For example, if $�(�)$ is $3{�}^2+2�-1$, the degree is 2.

2. Degree of Denominator $�(�)$: Denoted as $�$, it is the highest power of $�$ in the denominator. For example, if $�(�)$ is ${�}^3-2�+4$, the degree is 3.

Step 3: Compare Degrees to Determine Asymptotes

The behavior of the rational function depends on the relationship between the degrees $�$ and $�$:

• If $�<�$, there is a horizontal asymptote at $�=0$ (the x-axis). As $�$ becomes very large in magnitude ($+\mathrm{\infty }$ or $-\mathrm{\infty }$), the function approaches zero.

• If $�=�$, there is a horizontal asymptote at $�=\frac{�}{�}$, where $�$ is the leading coefficient of the numerator $�(�)$ and $�$ is the leading coefficient of the denominator $�(�)$. In this case, the function levels off at a constant value as $�$ becomes very large.

• If $�>�$, there is no horizontal asymptote. The function does not level off at a constant value as $�$ becomes very large; instead, it grows without bound or exhibits oscillations, depending on the specific function.

Step 4: Interpret the Asymptotes

• A horizontal asymptote at $�=0$ indicates that the function approaches zero as $�$ becomes very large.

• A horizontal asymptote at a specific constant value $�=\frac{�}{�}$ indicates that the function approaches that constant value as $�$ becomes very large.

• The absence of a horizontal asymptote suggests that the function's behavior is not bounded by a constant as $�$ approaches $\pm \mathrm{\infty }$.

Step 5: Sketch the Graph

Once you've identified the horizontal asymptotes and their locations, you can sketch the graph of the rational function, including the horizontal asymptotes as horizontal lines on the graph.

Examples:

1. For the rational function $�(�)=\frac{3{�}^2}{{�}^2+1}$, $�=2$ and $�=2$, so there is a horizontal asymptote at $�=\frac{3}{1}=3$.

2. For the rational function $�(�)=\frac{{�}^3-2�+4}{3{�}^2-1}$, $�=3$ and $�=2$, so there is no horizontal asymptote.

Identifying horizontal asymptotes is a fundamental aspect of analyzing rational functions, as they provide insights into the function's behavior as $�$ approaches infinity or negative infinity.

Horizontal asymptotes are important features of rational functions that describe the behavior of the function as the independent variable ($�$) approaches positive or negative infinity. They provide information about where the function levels off as $�$ becomes very large or very small. The presence and location of horizontal asymptotes depend on the degrees of the polynomials in the numerator ($�(�)$) and denominator ($�(�)$) of the rational function $�(�)=\frac{�(�)}{�(�)}$.

Here are the key principles for identifying horizontal asymptotes of rational functions:

1. Compare the Degrees of the Polynomials:

• Degree of Numerator $�(�)$: Denoted as $�$, it is the highest power of $�$ in the numerator. For example, if $�(�)$ is $3{�}^2+2�-1$, the degree is 2 ($�=2$).
• Degree of Denominator $�(�)$: Denoted as $�$, it is the highest power of $�$ in the denominator. For example, if $�(�)$ is ${�}^3-2�+4$, the degree is 3 ($�=3$).

2. Compare the Degrees to Determine Asymptotes:

• If $�<�$, there is a horizontal asymptote at $�=0$ (the x-axis). As $�$ becomes very large in magnitude ($+\mathrm{\infty }$ or $-\mathrm{\infty }$), the function approaches zero.
• If $�=�$, there is a horizontal asymptote at $�=\frac{�}{�}$, where $�$ is the leading coefficient of the numerator $�(�)$ and $�$ is the leading coefficient of the denominator $�(�)$. In this case, the function levels off at a constant value as $�$ becomes very large.
• If $�>�$, there is no horizontal asymptote. The function does not level off at a constant value as $�$ becomes very large; instead, it grows without bound or exhibits oscillations, depending on the specific function.

3. Interpret the Asymptotes:

• A horizontal asymptote at $�=0$ indicates that the function approaches zero as $�$ becomes very large.
• A horizontal asymptote at a specific constant value $�=\frac{�}{�}$ indicates that the function approaches that constant value as $�$ becomes very large.
• The absence of a horizontal asymptote suggests that the function's behavior is not bounded by a constant as $�$ approaches $\pm \mathrm{\infty }$.

4. Sketch the Graph:

• Once you've identified the horizontal asymptotes and their locations, you can sketch the graph of the rational function, including the horizontal asymptotes as horizontal lines on the graph.

Here are examples to illustrate the identification of horizontal asymptotes:

Example 1: For the rational function $�(�)=\frac{2{�}^3-4{�}^2+3}{{�}^3+�-2}$:

• Degree of Numerator: $�=3$
• Degree of Denominator: $�=3$
• Since $�=�$, there is a horizontal asymptote at $�=\frac{2}{1}=2$.

Example 2: For the rational function $�(�)=\frac{4{�}^2-6�+1}{{�}^3-2�+4}$:

• Degree of Numerator: $�=2$
• Degree of Denominator: $�=3$
• Since $�<�$, there is a horizontal asymptote at $�=0$.

Example 3: For the rational function $ℎ(�)=\frac{3{�}^4+2{�}^3-5}{2{�}^2-�+1}$:

• Degree of Numerator: $�=4$
• Degree of Denominator: $�=2$
• Since $�>�$, there is no horizontal asymptote.

Identifying horizontal asymptotes helps in understanding the long-term behavior of rational functions and is an essential part of their analysis.

Intercepts of rational functions, like any other types of functions, include both x-intercepts and y-intercepts. X-intercepts are the points where the graph of the function crosses the x-axis, and y-intercepts are the points where the graph crosses the y-axis. To find these intercepts for a rational function, follow these steps:

1. X-Intercepts (Zeros):

X-intercepts are the points where the graph of the rational function crosses the x-axis. They correspond to the values of $�$ for which the function $�(�)$ equals zero ($�(�)=0$). To find the x-intercepts, solve the equation $�(�)=0$ for $�$.

Here's how to find x-intercepts:

• Set the numerator of the rational function equal to zero: $�(�)=0$.
• Solve for $�$ to find the zeros of the numerator.
• These zeros represent the x-intercepts of the rational function.

2. Y-Intercept:

The y-intercept is the point where the graph of the rational function crosses the y-axis. It corresponds to the value of $�(�)$ when $�=0$. To find the y-intercept, substitute $�=0$ into the rational function and evaluate $�(0)$.

Here's how to find the y-intercept:

• Set $�=0$ in the rational function: $�(0)=\frac{�(0)}{�(0)}$.
• Evaluate $�(0)$ to find the y-intercept.

Here are some examples to illustrate finding intercepts of rational functions:

Example 1: Finding X-Intercepts and Y-Intercepts

Consider the rational function $�(�)=\frac{{�}^2-4}{�-2}$.

1. X-Intercepts (Zeros):

   • Set the numerator equal to zero: ${�}^2-4=0$.
   • Solve for $�$: ${�}^2-4=(�-2)(�+2)=0$.
   • Zeros: $�=2$ and $�=-2$.

2. Y-Intercept:

   • Set $�=0$ in the rational function: $�(0)=\frac{0^2-4}{0-2}$.
   • Evaluate $�(0)$: $�(0)=\frac{-4}{-2}=2$.
   • The y-intercept is at the point (0, 2).

Example 2: Finding X-Intercepts and Y-Intercepts

Consider the rational function $�(�)=\frac{3{�}^3-6{�}^2}{2{�}^2-4�}$.

1. X-Intercepts (Zeros):

   • Set the numerator equal to zero: $3{�}^3-6{�}^2=0$.
   • Factor out common terms: $3{�}^2(�-2)=0$.
   • Zeros: $�=0$ and $�=2$.

2. Y-Intercept:

   • Set $�=0$ in the rational function: $�(0)=\frac{3(0{)}^3-6(0{)}^2}{2(0{)}^2-4(0)}$.
   • Evaluate $�(0)$: $�(0)=\frac{0}{0}=$ undefined.
   • In this case, there is no y-intercept because the function is undefined at $�=0$.

Finding intercepts is a fundamental step in analyzing the behavior of rational functions and understanding where their graphs intersect with the coordinate axe.

Graphing rational functions involves visualizing the behavior of the function on a coordinate plane. Rational functions, which are ratios of two polynomials, can exhibit various features such as vertical asymptotes, horizontal asymptotes, x-intercepts, y-intercepts, and holes (removable discontinuities). Here's a step-by-step guide on how to graph a rational function:

1. Find Vertical Asymptotes (VA):

• Vertical asymptotes are vertical lines that the graph approaches but never crosses. They occur where the denominator of the rational function equals zero ($�(�)=0$).
• Solve $�(�)=0$ to find the values of $�$ where vertical asymptotes exist.

2. Find Horizontal Asymptotes (HA):

• Horizontal asymptotes are horizontal lines that the graph approaches as $�$ becomes very large or very small. They depend on the degrees of the numerator and denominator polynomials.
• Compare the degrees of $�(�)$ (numerator) and $�(�)$ (denominator) to determine horizontal asymptotes:
  • If $�<�$, the horizontal asymptote is $�=0$.
  • If $�=�$, the horizontal asymptote is $�=\frac{�}{�}$ (leading coefficients of $�(�)$ and $�(�)$).
  • If $�>�$, there is no horizontal asymptote.

3. Find x-Intercepts (Zeros):

• x-Intercepts are where the graph crosses the x-axis ($�=0$). They occur where the numerator equals zero ($�(�)=0$).
• Solve $�(�)=0$ to find the x-intercepts.

4. Find y-Intercepts:

• y-Intercepts are where the graph crosses the y-axis ($�=0$). To find the y-intercept, substitute $�=0$ into the rational function and solve for $�$.

5. Determine the Behavior Near Vertical Asymptotes:

• Determine the behavior of the function as $�$ approaches the vertical asymptotes. You can use limits to understand whether the function approaches positive or negative infinity or converges to a finite value.

6. Identify Holes (Removable Discontinuities):

• Holes occur when there are common factors in the numerator and denominator that cancel out at a specific point, resulting in a gap in the graph.
• Find these points of discontinuity by solving for $�$ where $�(�)$ and $�(�)$ have common factors. Then, simplify the function by canceling out the common factors.

7. Sketch the Graph:

• Use all the information obtained in the previous steps to sketch the graph of the rational function on a coordinate plane.
• Draw the vertical asymptotes as dashed vertical lines, and the horizontal asymptotes as dashed horizontal lines.
• Plot the x-intercepts, y-intercepts, and any holes.
• Connect the points to create a smooth curve that approaches the asymptotes.

8. Label Key Points:

• Label the key points, such as the coordinates of x-intercepts, y-intercepts, and any points of interest.
• Provide arrows indicating the behavior of the function as $�$ approaches the asymptotes.

9. Analyze and Verify:

• Analyze the graph to verify that it matches the expected behavior based on the calculations of asymptotes, intercepts, and discontinuities.

Remember that graphing rational functions can be more complex for functions with higher degrees or multiple terms. In such cases, it may be helpful to use graphing software or calculators to visualize the graph accurately.

Writing rational functions involves creating mathematical expressions that are represented as a ratio of two polynomial expressions. These rational functions can describe various real-world phenomena and mathematical relationships. Here's how to write rational functions and some examples:

1. Basic Form: A rational function is typically written in the form:

   $�(�)=\frac{�(�)}{�(�)}$

   Where:

   • $�(�)$ is the rational function.
   • $�(�)$ is the numerator polynomial, which can be any polynomial expression.
   • $�(�)$ is the denominator polynomial, which can also be any polynomial expression, with the condition that $�(�)\mathrm{\ne }0$ for any $�$ in the domain.

2. Domain: Determine the domain of the rational function. The domain consists of all values of $�$ for which the denominator $�(�)$ is not equal to zero. You must exclude any values of $�$ that make the denominator zero from the domain.

3. Simplify: Simplify the rational function by factoring the numerator and denominator and canceling common factors if possible. This simplification can help you analyze the function more easily.

4. Examples:

   a. Simple Rational Function:

   $�(�)=\frac{2�+1}{�-3}$

   • Here, $�(�)=2�+1$ and $�(�)=�-3$.
   • The domain is all real numbers except $�=3$ because $�(3)=0$.

   b. Complex Rational Function:

   $�(�)=\frac{{�}^2-4}{{�}^3+2{�}^2-8�}$

   • Here, $�(�)={�}^2-4$ and $�(�)={�}^3+2{�}^2-8�$.
   • The domain is all real numbers except $�=0$ because $�(0)=0$.

5. Vertical and Horizontal Asymptotes: Analyze the function to find any vertical and horizontal asymptotes. Vertical asymptotes occur where the denominator is zero, and horizontal asymptotes represent the behavior of the function as $�$ approaches positive or negative infinity.

6. Graphing: You can graph rational functions using the information obtained above. Plot key points, such as the x-intercepts, y-intercepts, vertical asymptotes, and any holes in the graph. Consider the behavior of the function near critical points.

7. Additional Features: Rational functions can have additional features like holes, slant asymptotes, and local extrema. These features can be explored further by analyzing the function and its derivatives.

8. Special Cases: Some special cases include improper rational functions where the degree of the numerator is equal to or greater than the degree of the denominator. In such cases, you may need to perform polynomial long division to simplify the function.

Overall, writing and analyzing rational functions involve understanding the properties of polynomials, factoring, and considering domain restrictions to describe mathematical relationships in a concise and accurate manner.

To write a rational function from its intercepts and asymptotes, you'll need to consider the information given by these key features and construct a rational expression that meets those criteria. Here's how you can do it:

1. Intercepts:

   • Start by identifying the x-intercepts (zeros) and y-intercepts of the function.
   • The x-intercepts are the values of $�$ where the function equals zero ($�(�)=0$), and the y-intercept is the value of $�(0)$.

2. Asymptotes:

   • Determine the vertical and horizontal asymptotes of the function.
   • Vertical asymptotes occur where the denominator of the rational function equals zero.
   • Horizontal asymptotes describe the behavior of the function as $�$ approaches positive or negative infinity. These can be horizontal lines with equations of the form $�=�$, where $�$ is a constant.

3. Construct the Rational Function:

   • Use the information from the intercepts and asymptotes to construct the rational function.
   • The x-intercepts correspond to factors in the numerator, while the vertical asymptotes correspond to factors in the denominator.
   • The y-intercept gives you information about the constant term in the numerator.
   • The horizontal asymptotes help determine the degree of the numerator and denominator.

4. Example:

   Let's say you have the following information for a rational function:

   • x-intercepts: $�=-2$ and $�=3$.
   • y-intercept: $�=-1$.
   • Vertical asymptote: $�=1$.
   • Horizontal asymptote: $�=2$.

   You can start constructing the rational function:

   • The x-intercepts indicate factors in the numerator: $(�+2)$ and $(�-3)$.
   • The y-intercept gives you the constant term in the numerator: $���������=-1$.
   • The vertical asymptote gives you a factor in the denominator: $(�-1)$.
   • The horizontal asymptote implies that the degrees of the numerator and denominator must be the same, so you can add ${�}^2$ to both the numerator and denominator:

   $�(�)=\frac{���������}{�����������}=\frac{(-1)(�+2)(�-3)}{(�-1)({�}^2)}$

   This is your rational function. You can simplify further if needed.

5. Simplify:

   • If possible, simplify the rational function by canceling any common factors between the numerator and denominator.
   • In this case, you can simplify further by canceling $(�-1)$ from both the numerator and denominator:

   $�(�)=\frac{(-1)(�+2)(�-3)}{(�-1)({�}^2)}=\frac{(-1)(�+2)(�-3)}{({�}^2)}$

That's how you can write a rational function based on intercepts and asymptotes. This process allows you to create a function that matches the given features while ensuring that the function is defined over the specified domain.

Let's work through a couple of examples of writing rational functions based on intercepts and asymptotes.

Example 1:

Suppose you are given the following information:

• x-intercepts: $�=-1$ and $�=2$.
• y-intercept: $�=4$.
• Vertical asymptote: $�=3$.
• Horizontal asymptote: $�=0$.

Now, let's construct the rational function:

1. The x-intercepts give us factors in the numerator: $(�+1)$ and $(�-2)$.
2. The y-intercept provides the constant term in the numerator: $���������=4$.
3. The vertical asymptote corresponds to a factor in the denominator: $(�-3)$.
4. The horizontal asymptote indicates that the degrees of the numerator and denominator should be the same. So, we can add ${�}^2$ to both the numerator and denominator.

The rational function becomes:

$�(�)=\frac{(4)(�+1)(�-2)}{(�-3)({�}^2)}$

You can further simplify this expression if needed.

Example 2:

Let's say you have this information:

• x-intercepts: $�=-3$ and $�=4$.
• y-intercept: $�=-2$.
• Vertical asymptote: $�=2$.
• Horizontal asymptote: $�=1$.

Construct the rational function as follows:

1. The x-intercepts translate to factors in the numerator: $(�+3)$ and $(�-4)$.
2. The y-intercept provides the constant term in the numerator: $���������=-2$.
3. The vertical asymptote corresponds to a factor in the denominator: $(�-2)$.
4. The horizontal asymptote implies that the degrees of the numerator and denominator should be the same. To achieve this, we can add ${�}^2$ to both the numerator and denominator.

The rational function becomes:

$�(�)=\frac{(-2)(�+3)(�-4)}{(�-2)({�}^2)}$

Again, you can simplify this expression further if necessary.

These are examples of how to write rational functions based on given intercepts and asymptotes. The key is to use the information about intercepts and asymptotes to construct the numerator and denominator of the rational function while considering the degrees and constants involved.