Question

1. the parent rational function ( y=\frac{1}{x} ) is undefined at ( x = 0 ). complete the table to evaluate this function for values of ( x ) very near 0. what do the values in the table indicate about the behavior of the graph?

( x )	( y=\frac{1}{x} )
-0.1	-10
-0.01
-0.001
0	undefined
0.001
0.01
0.1
1

agile

Explanation:

Step1: Evaluate \( y = \frac{1}{x} \) for \( x=-0.01 \)

Substitute \( x = - 0.01 \) into \( y=\frac{1}{x} \), we get \( y=\frac{1}{-0.01}=-100 \)

Step2: Evaluate \( y = \frac{1}{x} \) for \( x=-0.001 \)

Substitute \( x=-0.001 \) into \( y = \frac{1}{x} \), we get \( y=\frac{1}{-0.001}=-1000 \)

Step3: Evaluate \( y = \frac{1}{x} \) for \( x = 0.001 \)

Substitute \( x = 0.001 \) into \( y=\frac{1}{x} \), we get \( y=\frac{1}{0.001}=1000 \)

Step4: Evaluate \( y = \frac{1}{x} \) for \( x = 0.01 \)

Substitute \( x = 0.01 \) into \( y=\frac{1}{x} \), we get \( y=\frac{1}{0.01}=100 \)

Step5: Evaluate \( y = \frac{1}{x} \) for \( x = 0.1 \)

Substitute \( x = 0.1 \) into \( y=\frac{1}{x} \), we get \( y=\frac{1}{0.1}=10 \)

Step6: Evaluate \( y = \frac{1}{x} \) for \( x = 1 \)

Substitute \( x = 1 \) into \( y=\frac{1}{x} \), we get \( y=\frac{1}{1}=1 \)

For the behavior of the graph:
As \( x\to0^{-}\) ( \( x \) approaches \( 0\) from the left - hand side), \( y=\frac{1}{x}\to-\infty\). As \( x\to0^{+}\) ( \( x \) approaches \( 0\) from the right - hand side), \( y = \frac{1}{x}\to+\infty\). This indicates that the line \( x = 0\) (the \( y\) - axis) is a vertical asymptote of the graph of \( y=\frac{1}{x}\)

Answer:

\( x\)	\( y=\frac{1}{x}\)
\(-0.1\)	\(-10\)
\(-0.01\)	\(-100\)
\(-0.001\)	\(-1000\)
\(0\)	Undefined
\(0.001\)	\(1000\)
\(0.01\)	\(100\)
\(0.1\)	\(10\)
\(1\)	\(1\)

The values in the table indicate that the line \( x = 0\) is a vertical asymptote of the graph of \( y=\frac{1}{x}\). As \( x\) approaches \( 0\) from the left (\(x<0,x\to0\)), \(y\) approaches \(-\infty\); as \( x\) approaches \( 0\) from the right (\(x > 0,x\to0\)), \(y\) approaches \(+\infty\)