Problem 1 - Algebra and Trigonometry

Question

A rational function is given. (a) Complete each table for the function. (b) Describe the behavior of the function nea its vertical asymptote, based on Tables 1 and \(2 .\) (c) Determine the horizontal asymptote, based on Tables 3 and \(4 .\) \(\begin{array}{|c|c|}\hline x & {r(x)} \\ \hline 1.5 & {} \\ {1.9} & {} \\\ {1.99} & {} & {} \\ {1.999} & {} \\ \hline\end{array}\) \(\begin{array}{|c|c|}\hline x & {r(x)} \\ \hline 2.5 & {} \\ {2.1} & {} & {} \\\ {2.01} & {} \\ {2.001} & {} \\ \hline\end{array}\) \(\begin{array}{|c|c|}\hline x & {r(x)} \\ \hline 10 & {} \\ {50} & {} \\\ {100} & {} \\ {1000} & {} \\ \hline\end{array}\) \(\begin{array}{|c|c|}\hline x & {r(x)} \\ \hline-10 & {} \\ {-50} & {} \\ {-100} & {} \\ {-1000} & {} \\\ \hline\end{array}\) \(r(x)=\frac{x}{x-2}\)

Step-by-Step Solution

Answer

Vertical asymptote is at \( x = 2 \); horizontal asymptote is \( y = 1 \).

Step 1: Fill Table 1

Calculate the values of the function \( r(x) = \frac{x}{x - 2} \) for the values given in Table 1. - For \( x = 1.5 \), \( r(1.5) = \frac{1.5}{1.5 - 2} = -3 \).- For \( x = 1.9 \), \( r(1.9) = \frac{1.9}{1.9 - 2} = -19 \).- For \( x = 1.99 \), \( r(1.99) = \frac{1.99}{1.99 - 2} = -199 \).- For \( x = 1.999 \), \( r(1.999) = \frac{1.999}{1.999 - 2} = -1999 \).

Step 2: Fill Table 2

Calculate the values of the function \( r(x) = \frac{x}{x - 2} \) for the values given in Table 2.- For \( x = 2.5 \), \( r(2.5) = \frac{2.5}{2.5 - 2} = 5 \).- For \( x = 2.1 \), \( r(2.1) = \frac{2.1}{2.1 - 2} = 21 \).- For \( x = 2.01 \), \( r(2.01) = \frac{2.01}{2.01 - 2} = 201 \).- For \( x = 2.001 \), \( r(2.001) = \frac{2.001}{2.001 - 2} = 2001 \).

Step 3: Behavior Near Vertical Asymptote

Based on the results from Tables 1 and 2, observe the behavior of \( r(x) \) as \( x \) approaches \( 2 \) from both sides. As \( x \to 2^- \), \( r(x) \to -\infty \). As \( x \to 2^+ \), \( r(x) \to \infty \). Thus, the vertical asymptote at \( x = 2 \) makes the function decrease to negative infinity as it approaches from the left and increase to positive infinity as it comes from the right.

Step 4: Fill Table 3

Calculate the values of the function \( r(x) = \frac{x}{x - 2} \) for large positive values given in Table 3.- For \( x = 10 \), \( r(10) = \frac{10}{8} = 1.25 \).- For \( x = 50 \), \( r(50) = \frac{50}{48} \approx 1.0417 \).- For \( x = 100 \), \( r(100) = \frac{100}{98} \approx 1.0204 \).- For \( x = 1000 \), \( r(1000) = \frac{1000}{998} \approx 1.002 \).

Step 5: Fill Table 4

Calculate the values of the function \( r(x) = \frac{x}{x - 2} \) for large negative values given in Table 4.- For \( x = -10 \), \( r(-10) = \frac{-10}{-12} = 0.8333 \).- For \( x = -50 \), \( r(-50) = \frac{-50}{-52} \approx 0.9615 \).- For \( x = -100 \), \( r(-100) = \frac{-100}{-102} \approx 0.9804 \).- For \( x = -1000 \), \( r(-1000) = \frac{-1000}{-1002} \approx 0.998 \).

Step 6: Determine the Horizontal Asymptote

Based on the results from Tables 3 and 4, observe the behavior of \( r(x) \) as \( x \) approaches positive and negative infinity. In both cases \( r(x) \to 1 \). Therefore, the horizontal asymptote of the function is \( y = 1 \).

Key Concepts

Vertical Asymptote

The vertical asymptote of a rational function is a line that the graph of the function approaches but never actually touches or crosses. This occurs when the denominator of the rational function is zero, which causes the function to be undefined at those points.
For the function given, \( r(x) = \frac{x}{x - 2} \), the denominator becomes zero when \( x = 2 \). Therefore, \( x = 2 \) is a vertical asymptote.
Let's explore what happens as the function approaches this point:

• As \( x \) moves closer to 2 from the left (values less than 2), the values of \( r(x) \) grow very large in the negative direction. This is because the denominator, \( x - 2 \), approaches zero and \( r(x) \) approaches negative infinity, which means \( r(x) \) decreases without bound.


• As \( x \) comes from the right (values greater than 2), \( r(x) \) increases to positive infinity. This behavior shows that \( r(x) \) grows very large in the positive direction as the denominator approaches zero.

This dramatic change from negative to positive infinity along the line \( x = 2 \) confirms the presence of the vertical asymptote there.

Horizontal Asymptote

The horizontal asymptote of a rational function indicates its end behavior as \( x \) approaches positive or negative infinity. It helps us understand the function's typical value at these extreme values of \( x \).
For the function \( r(x) = \frac{x}{x - 2} \), we must examine what happens as \( x \) increases or decreases without bound.

Approaching Positive Infinity

• When \( x \) is a large positive number, the function \( r(x) = \frac{x}{x - 2} \) simplifies to about \( \frac{x}{x} = 1 \), since the -2 becomes insignificant in comparison with \( x \). From Tables 3, as \( x \) grows larger, \( r(x) \) approaches the value 1.

Approaching Negative Infinity

• Similarly, for large negative values of \( x \), \( r(x) \) approaches \( 1 \) as the magnitude of \( x \) dominates over the constant -2 in the denominator, evident from Table 4 results.

Thus, whether \( x \) goes to positive or negative infinity, \( r(x) \) approaches \( y = 1 \) consistently. Therefore, the horizontal asymptote of this function is \( y = 1 \).

Function Behavior

Function behavior near asymptotes dictates how a function reacts around its undefined points and as it stretches to infinity. Understanding this behavior is essential for sketching and comprehending the overall character of rational functions.

Behavior Near Vertical Asymptote

• As \( x \to 2^- \) (approaching 2 from the left), \( r(x) \), according to our observations, dives down towards negative infinity, explaining a sharp downtrend on the graph.


• Conversely, as \( x \to 2^+ \), the function value shoots up to positive infinity, highlighting a steep ascent.

End Behavior

The end behavior for \( r(x) = \frac{x}{x - 2} \) shows that:

• For \( x \to \infty \), \( r(x) \to 1 \), indicating that as \( x \) grows very positive, the values stabilize around the horizontal asymptote \( y = 1 \).


• For \( x \to -\infty \), \( r(x) \to 1 \), aligning with the same pattern on the negative end.

This implies that at the extremes, the graph tends toward a steady level close to \( y = 1 \), defining a flat, horizontal behavior far away from the vertical asymptote's perturbations.