Dominio rango y grafico función Racional | Caso 2 ejemplo 1

Understanding Rational Functions: Domain and Range

Introduction to the Function

• The video begins with an overview of finding the domain, range, and graphing a rational function where the degree of the denominator is greater than that of the numerator.

Degrees of Numerator and Denominator

• The degree of the numerator is identified as 1 (the exponent of x), while the degree of the denominator is 2 (the maximum exponent). This establishes that the numerator's degree is less than that of the denominator.

Steps for Graphing

• The presenter outlines four steps for graphing: factor expressions if possible, find vertical asymptotes to determine domain, identify horizontal asymptotes, and locate zeros for additional points on the graph.

Vertical Asymptotes and Domain

• Vertical asymptotes are determined by setting factors in the denominator equal to zero. The domain consists of all real numbers except these asymptote values (-1 and 1).

Horizontal Asymptote Explanation

• Since the degree of the numerator is less than that of the denominator, there will always be a horizontal asymptote at y = 0. This simplifies understanding without complex calculations.

Factoring Expressions

Identifying Factors

• The expression in this case cannot be factored easily since it’s linear in nature. However, factoring helps simplify functions when applicable.

Finding Vertical Asymptotes

• To find vertical asymptotes, set each factor from a factored form equal to zero. This leads to two solutions: x = -1 and x = 1.

Establishing Domain

Writing Domain

• The domain includes all real numbers except for x = -1 and x = 1 due to vertical asymptotes.

Graphing Asymptotes

Visual Representation

• Both vertical lines representing asymptotes are drawn on a Cartesian plane at x = -1 and x = 1; these lines indicate where the function does not exist.

Analyzing Horizontal Asymptote

Confirming Horizontal Asymptote

• A horizontal line at y = 0 is established based on previous findings about degrees; this indicates behavior as x approaches infinity or negative infinity.

Finding Zeros

Evaluating Function Behavior

• To find zeros, evaluate what happens when y equals zero or when substituting specific values into x; this helps understand how functions behave around critical points like (0,0).

Selecting Additional Points for Graphing

Choosing Points Near Asymptotes

• Select points close to vertical asymptotes (-3, -2), ensuring they provide insight into function behavior across different sections divided by these lines.

Calculating Function Values

Substituting Values into Function

• Substitute chosen values back into original equations to calculate corresponding y-values; this step aids in plotting accurate points on graphs.

Verifying Point Locations

Plotting Points Accurately

• After calculating various points such as (-2,-0.6), ensure they align correctly with expected behaviors near both horizontal and vertical asymptotes during graph construction.

Finalizing Graph Shape

Observations About Function Behavior

• Analyze how selected points influence overall shape; confirm whether they approach positive or negative infinities based on their proximity to defined asymptotic boundaries.

Conclusion on Ranges

Determining Overall Range

• Conclude that range encompasses all real numbers based on visual inspection from plotted graphs rather than numerical calculation alone; thus confirming comprehensive understanding through graphical representation.