Dominio rango y grafico función Racional | Caso 1 ejemplo 1

How to Find the Domain and Range of a Rational Function

Introduction to Rational Functions

• The video introduces the concept of finding the domain and range of rational functions, emphasizing the importance of understanding different cases based on the exponents in the function.

• It mentions three specific cases regarding the exponents: when they are equal, when the exponent in the numerator is greater than that in the denominator, and vice versa.

Vertical Asymptotes

• The speaker explains that vertical asymptotes are crucial for graphing rational functions. They suggest starting with these before plotting other points.

• A reminder is given about how to find vertical asymptotes by ensuring that denominators do not equal zero during division operations.

Finding Vertical Asymptotes

• The process begins by identifying where the denominator equals zero since division by zero is undefined.

• For example, if we have a function like x/3 , setting x - 3 = 0 helps find where vertical asymptotes occur.

Graphing Vertical Asymptotes

• Once identified, vertical asymptotes can be graphed as straight lines. In this case, an asymptote occurs at x = 3 .

• Understanding where these lines lie helps predict how the function behaves near those values.

Horizontal Asymptotes

• The discussion shifts to horizontal asymptotes, which depend on comparing maximum exponents in both numerator and denominator.

• When both maximum exponents are equal (in this case both being 1), a formula is provided: y = fractextleading coefficient of numeratortextleading coefficient of denominator .

Calculating Horizontal Asymptote Coefficients

Horizontal Asymptotes and Finding Zeros in Functions

Understanding Horizontal Asymptotes

• The horizontal asymptote is identified as y = 2/1 , which simplifies to y = 2 . This line represents the behavior of the graph as it approaches infinity.

• The significance of asymptotes lies in understanding that the graph will approach these lines but never touch them, indicating where the function behaves at extreme values.

Steps to Find Zeros of a Function

• Finding zeros is not mandatory but recommended for better graphing. It involves determining points where either x = 0 or y = 0 .

• To find when x = 0 , substitute into the function: if y = 2x - 5/(x - 3) , then substituting gives a value of y = 5/3 .

Identifying Points on the Graph

• The point found when x = 0 is (0, 5/3 ), marking an intersection with the Y-axis.

• When finding where y = 0, rearranging leads to solving for x: this results in another point at (2.5, 0).

Creating a Value Table for Graphing

• A table of values helps visualize how the function behaves around its asymptote and other critical points.

• Important values include those near vertical asymptotes; here, when x = 3, it indicates undefined behavior since division by zero occurs.

Evaluating Additional Points

• Choosing values like (1, 2, and others around the asymptote helps clarify how the graph behaves on both sides.

Graphing Rational Functions and Understanding Asymptotes

Finding Points on the Graph

• The process begins by substituting x = 1 into the equation, resulting in y = 3/2 . This is expressed as a decimal (1.5), which is preferred by many students.

• Next, when x = 2 , substituting gives y = 1 . Students are encouraged to practice this substitution themselves.

• For x = 4 , the calculation yields y = 3 . This reinforces the importance of practicing substitutions for different values of x .

• When substituting x = 5 , it results in y = 2.5 . These points are crucial for plotting the graph accurately.

• The identified points include (0, 1.6), (1, 1.5), (2, 1), and (4, 3). The speaker emphasizes that these points will help determine the overall shape of the graph.

Analyzing Asymptotes

• The function approaches an asymptote as it extends towards negative infinity. This indicates that understanding asymptotic behavior is essential for graphing rational functions.

• Observations about how the graph behaves near asymptotes suggest that it remains below a certain threshold due to its structure.

• A more refined version of the graph is presented after initial plotting, highlighting key features such as proximity to asymptotes and overall shape.

Domain and Range Considerations

• The domain includes all real numbers except where vertical asymptotes occur. This means identifying restrictions based on undefined values in the function.

• Similarly, the range encompasses all real numbers but excludes horizontal asymptotes. Understanding these exclusions helps clarify function behavior across its entire span.

General Rules for Rational Functions

• When both numerator and denominator have equal exponents, specific rules apply regarding domain and range:

• Domain: All reals minus vertical asymptote.

• Range: All reals minus horizontal asymptote value.

• The speaker concludes with a summary of findings related to domains and ranges based on their observations from earlier calculations.

Practice Problems

• Viewers are encouraged to practice similar problems involving rational functions with equal maximum exponents in both numerator and denominator to reinforce learning concepts discussed throughout the video.

• Emphasis is placed on recognizing rational functions through their structure—specifically noting coefficients associated with maximum exponents during analysis of vertical and horizontal asymptotes.

Graphing Rational Functions

Finding Points on the Graph

• The discussion begins with calculating points for a rational function, where if x = 0 , then y = 5/4 or approximately 1.2.

• The vertical asymptote is identified at x = 2 . This point serves as a central reference for graphing, indicating that there are no values of the function at this point.

• By substituting different values for x , additional points are calculated:

• When x = 1 , y = 2 .

• When x = 3 , y = -1 .

• When x = 4 , y = -1/4 .

Asymptotes and Graph Behavior

• The vertical asymptote at x = 2 and horizontal asymptote at y = 1/2 are crucial in understanding the behavior of the graph as it approaches these lines.

• Key points plotted include (0, 1.25), (1, 2), (3, -1), and (4, -0.25). The graph will approach both asymptotes as it tends to infinity.

Conclusion and Further Learning