Problem 24 - College Algebra and Calculus: An Applied Approach

Question

Use a graphing utility to graph the function. Describe the shape of the graph for very large and very small values of \(x\). (a) \(f(x)=\frac{8}{1+e^{-0.5 x}}\) (b) \(g(x)=\frac{8}{1+e^{-0.5 / x}}\)

Step-by-Step Solution

Answer

For the first function, as \(x\to -\infty\), \(f(x)\to 0\) and as \(x\to \infty\), \(f(x)\to 8\). For the second function, \(\lim_{{x \to 0}} g(x) = \infty\) and as \(x\) gets large or small, \(g(x)\to 0\).

Step 1: Plot the First Function

Using the graphing tool, enter the following function: \(f(x)=\frac{8}{1+e^{-0.5 x}}\). After generating the graph, observe its shape and how it changes for large and small values of \(x\).

Step 2: Describe Behavior of the First Function

As \(x\) approaches negative infinity (\(x\to -\infty\)), the value of \(f(x)\) approaches 0; the graph gets closer and closer to the x-axis but never touch it. As \(x\) approaches positive infinity (\(x\to \infty\)), \(f(x)\) approaches the value 8. It means the graph gets closer and closer to the line \(y=8\) as \(x\) becomes larger and larger.

Step 3: Plot the Second Function

Enter the second function \(g(x)=\frac{8}{1+e^{-0.5 / x}}\) into the graphing tool. Again, observe the shape for both high and low \(x\) values.

Step 4: Describe Behavior of the Second Function

For extremely large or small values of \(x\), the function \(g(x)\) approaches the value 0. However, around \(x=0\), \(g(x)\) aggressively increases and \(\lim_{{x \to 0}} g(x) = \infty\).