Question

In Exercises \(75-86\), use a computer algebra system to analyze the graph of the function. Label any extrema and/or asymptotes that exist. $$ g(x)=\sin \left(\frac{x}{x-2}\right), \quad x>3 $$

Short answer

The function \( g(x) = \sin \left(\frac{x}{x-2}\right) \) for \( x>3 \) has a cyclic behavior without global extrema due to its sine structure. Local extrema may exist depending on the domain \( x>3 \). There is no asymptote within the given domain.

Step-by-step solution

Plotting the function

Begin by using a computer algebra system to plot the graph \( g(x) = \sin \left(\frac{x}{x-2}\right) \) for the domain \( x>3 \). Be sure to enlarge the view of the graph to clearly see the behavior of the function and find any extrema or asymptotes.

Locating the extrema

Extrema are the points of local maximum and minimum values of a function. There are no global extrema (minimum or maximum points) for the function \( g(x) = \sin \left(\frac{x}{x-2}\right) \) because it is a sine function which oscillates between -1 and 1 indefinitely. Identify if there are any local extrema in the given domain \( x>3 \)

Identifying the asymptotes

Asymptotes are lines that the graph approaches but never touches. For this function, a vertical asymptote exists at \( x=2 \) because the fraction in the argument of the sine function becomes undefined at this point. However, because we are only considering \( x>3 \), this point does not fall in the domain of our function. Therefore, no asymptotes occur in the given domain \( x > 3 \)