Consider the rational function f(x)=(1)/(x). a) Determine...

Consider the rational function $f(x)=\frac{1}{x}$. a) Determine the domain of the function. b) Complete the following table for the function. $$ \begin{array}{l|l|l|l|l|l|l|l|l|l|l} \hline x & -10 & -1 & -0.5 & -0.1 & -0.01 & 0.01 & 0.1 & 0.5 & 1 & 10 \\ \hline y & & & & & & & & & & \\ \hline \end{array} $$ c) Draw the graph of $f(x)=\frac{1}{x}$. Consider what happens to the function as $x$ gets closer and closer to 0 , approaching 0 from both the left and right sides. d) Can this fraction ever have a value of 0? Explain your answer.

Consider the rational function $f(x)=\frac{1}{x}$.
a) Determine the domain of the function.
b) Complete the following table for the function.
$$
\begin{array}{l|l|l|l|l|l|l|l|l|l|l}
\hline x & -10 & -1 & -0.5 & -0.1 & -0.01 & 0.01 & 0.1 & 0.5 & 1 & 10 \\
\hline y & & & & & & & & & & \\
\hline
\end{array}
$$
c) Draw the graph of $f(x)=\frac{1}{x}$. Consider what happens to the function as $x$ gets closer and closer to 0 , approaching 0 from both the left and right sides.
d) Can this fraction ever have a value of 0? Explain your answer.

Key Concepts

Graphical Interpretation

Graphing a rational function involves plotting points, understanding asymptotic behaviors, and highlighting regions of undefined values. This holistic view helps in sketching an accurate graph and comprehending the overall behavior of the function.

Limit Behavior

Analyzing the limit behavior of a function as it approaches specific values, especially near discontinuities, helps in understanding the function's tendencies such as approaching infinity or settling into a constant value. This is crucial in explaining the function's behavior near its vertical asymptotes.

Vertical Asymptotes

Vertical asymptotes occur where the function's denominator is zero (and the numerator is nonzero), leading the function to exhibit unbounded behavior. They are essential in understanding how the function behaves near points of discontinuity.

Domain of Rational Functions

The domain of a rational function includes all real numbers except those that make the denominator equal to zero. This concept is critical as division by zero is undefined, and identifying these restrictions is the first step in analyzing any rational function.

Horizontal Asymptotes

Horizontal asymptotes describe the end behavior of a function as the independent variable approaches positive or negative infinity. For many rational functions, including those similar to 1/x, the function approaches a constant value, indicating a horizontal asymptote.

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