Question

Plot the graph of the function \(f\) in an appropriate viewing window. (Note: The answer is not unique.) $$ f(x)=\frac{5 x}{x-1}+5 x $$

Short answer

The given function is a rational function, \(f(x) = \frac{5x}{x-1} + 5x\). To plot the graph, we find the key features: 1. x-intercept: Setting \(f(x) = 0\) gives an x-intercept at \(x = 0\). 2. y-intercept: Setting \(x = 0\) gives a y-intercept at \(f(0) = 0\). 3. Vertical asymptote: The denominator is zero when \(x = 1\). Choose an appropriate viewing window, for example, x-axis range -10 to 10 and y-axis range -10 to 10. Plot the graph of the function using a graphing tool, making sure to include the x-intercept at \(x = 0\), the y-intercept at \(f(0) = 0\), and the vertical asymptote at \(x = 1\).

Step-by-step solution

Identify the function type

The given function is a rational function since it has the form of a fraction. Also, note that the denominator and the numerator are both linear functions, which means that the graph will have the general shape of a rational function.

Find the intercepts

To find the x-intercepts (if any), set f(x) = 0 and solve for x: \[ 0 = \frac{5x}{x - 1} + 5x \] To find the y-intercepts (if any), set x = 0 and solve for f(x): \[ f(0) = \frac{5(0)}{0 - 1} + 5(0) \]

Identify vertical asymptotes

A rational function has a vertical asymptote when the denominator is equal to zero and the function is undefined. For the given function, the denominator will be zero when: \[ x - 1 = 0 \]

Choose an appropriate viewing window

Based on the information gained in Steps 2 and 3, we can choose an appropriate viewing window for our graph. An appropriate window will show the x-intercepts, the y-intercepts, and the vertical asymptotes. Since the answer is not unique, you may choose any window that clearly shows these features. For example, this viewing window can be used: - x-axis range: -10 to 10 - y-axis range: -10 to 10

Plot the graph

Using the information gathered during the previous steps and the chosen viewing window, plot the graph of the function: \[ f(x) = \frac{5x}{x - 1} + 5x \] Make sure to include the x-intercepts, y-intercepts, and vertical asymptotes in the graph. Remember that the graph can be plotted using a graphing tool or calculator, which will help visualize the function within the given window.