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Chapter 3: Problem 72

Use a graphing utility to graph the function. Be sure to use an appropriate viewing window. \(f(x)=3 \ln x-1\)

Short Answer

Expert verified
For the function \(f(x)=3 \ln x-1\), choose a viewing window that starts from an x-value close to 0 upward and extends from -3 to 3 on the y-axis. The graph should be an increasing curve due to the logarithmic nature, steeper because of the multiplier 3, and shifted down by one unit due to the -1.

Step by step solution

01

Understand the Function

First, understand the behavior of the function \(f(x)=3 \ln x-1\). This is a logarithmic function to the base e (natural log) scaled by a factor of 3 and then shifted down by 1. Logarithmic functions have a vertical asymptote at x=0 and are always increasing. The 3 multiplier will make the function steeper, and the -1 will shift the entire graph down by 1 unit.
02

Set an Appropriate Viewing Window

Since the function has a vertical asymptote at x=0, choose a window that starts from an x-value close to 0 (for example, 0.1) and goes to another positive value like x=5. This range should capture the key features of the logarithmic function. For the y-values, considering the down shift of 1, a range from -3 to 3 should provide a good overview of the function's behavior.
03

Graph the function

Input the function \(f(x)=3 \ln x-1\) into your graphing utility. Use the viewing window determined in Step 2. Your graph should resemble an increasing logarithmic curve, steeper due to the presence of the 3 multiplier, and shifted down by 1.

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