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) \(g(x)=\frac{8}{1+e^{-0.5 / x}}\)

Short 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-by-step solution

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 \(x\).

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.

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.

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 \(\lim_{{x \to 0}} g(x) = \infty\).