Problem 38 - Precalculus with Limits

Question

Use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. $$ k(x)=1 /(x-3) $$

Step-by-Step Solution

Answer

The graph of the function \(k(x)=1/(x-3)\) with the viewing window centered around x=3, shows a vertical asymptote at x=3 with the function approaching negative infinity for x<3, and approaching positive infinity for x>3.

Step 1: Understand the Function

The function \(k(x) = 1/(x-3)\) is a rational function. It is undefined at x=3, indicating that the graph will have a vertical asymptote at x=3. Thus, the domain of this function is all real numbers except 3.

Step 2: Choose the Viewing Window

An ideal viewing window for this function would be one that shows the behaviour of the function around the point x=3, where it is undefined. Hence, choose a range for x that centers around 3 and a range for y that will show the graph's broad behavior.

Step 3: Graph the Function

Using the selected viewing window, plot the function \(k(x)=1/(x-3)\) in the graphing utility. The graph shows a vertical asymptote at x=3 and the function approaches to this asymptote from both sides, from negative infinity for x<3 and positive infinity for x>3. This is characteristic of the function \(k(x)=1/(x-a)\)