Solve. For Exercises 83 and 84, see the second Concept Check...

Solve. For Exercises 83 and $84,$ see the second Concept Check in this section, for Exercises 85 and $86,$ see the third Concept Check. The domain of the function $f(x)=\frac{1}{x}$ is all real numbers except $0 .$ This means that the graph of this function will be in two pieces: one piece corresponding to $x$ values less than 0 and one piece corresponding to $x$ values greater than 0. Graph the function by completing the following tables, separately plotting the points, and connecting each set of plotted points with a smooth curve. (TABLE CANNOT COPY)

Solve. For Exercises 83 and $84,$ see the second Concept Check in this section, for Exercises 85 and $86,$ see the third Concept Check.
The domain of the function $f(x)=\frac{1}{x}$ is all real numbers except $0 .$ This means that the graph of this function will be in two pieces: one piece corresponding to $x$ values less than 0 and one piece corresponding to $x$ values greater than
0. Graph the function by completing the following tables, separately plotting the points, and connecting each set of plotted points with a smooth curve.
(TABLE CANNOT COPY)

Key Concepts

Domain of a Function

The domain refers to the set of all possible input values (usually x-values) for which the function is defined. Understanding the domain is crucial because it identifies any restrictions, such as values that would make a denominator zero or result in undefined expressions, thereby guiding the proper graphing and analysis of the function.

Rational Functions

Rational functions are quotients of polynomials and often have features such as discontinuities, asymptotes, and defined behavior in separate intervals. Analyzing rational functions typically involves understanding these characteristics, which aids in accurately determining and sketching their graphs.

Asymptotic Behavior

Asymptotes are lines that the graph of a function approaches but never actually touches. In the context of rational functions, vertical asymptotes occur where the function is undefined (often from a zero in the denominator), and horizontal asymptotes describe the end behavior as the absolute value of x increases. Recognizing asymptotic behavior helps in drawing the overall shape and limits of the graph.

Graphing from Tabulated Values

Using a table of values is a fundamental technique to graph a function. By computing the function’s output at selected input values, one can identify trends, discontinuities, and significant features of the graph. This method is particularly helpful for understanding piecewise behavior and the impact of domain restrictions on the overall graph of the function.

Transcript

Please give Ace some feedback