Question

In Exercises, use a graphing utility to graph the function. Determine any asymptotes of the graph. $$ g(x)=\frac{8}{1+e^{-0.5 / x}} $$

Short answer

The graph of the function can be plotted using a graphing utility. By observing the behavior of the function as \(x\) approaches positive and negative infinity, the horizontal asymptotes can be found. For the given function, it doesn't appear to have any vertical asymptotes as there are no values of \(x\) for which the function will be undefined.

Step-by-step solution

Graph the Function

To graph this function, a graphing utility such as Desmos or Geogebra can be used. Simply input the function \( g(x) = \frac{8}{1 + e^{-0.5/x}} \) into the graphing utility and plot the function.

Analyze the Graph

Once the function has been plotted, the behavior of the graph can be analyzed especially as \(x\) approaches positive and negative infinity. By observation, an expected behavior of the function can be ascertained, as it seems to be leveling off or approaching certain values as \(x\) becomes very large or very small.

Determine Asymptotes

Looking at the behavior of the function as \(x\) approaches positive or negative infinity, the vales that the function is approaching can be determined. These will be the horizontal asymptotes of the function. For the vertical asymptotes, which are the values at which the function would be undefined, for this function there doesn't appear to be any as the denominator of the function is never equal to zero.

Key concepts

Asymptotes

In mathematics, asymptotes are important concepts that help us understand the behavior of functions as they extend towards infinity or approach a particular point. Imagine an asymptote as a line that a function's graph gets infinitely close to, but never quite touches or crosses. Asymptotes can be encountered in the form of horizontal, vertical, or oblique/slant lines.

- **Horizontal asymptotes** indicate the behavior of a function as it heads towards positive or negative infinity on the x-axis. - **Vertical asymptotes** indicate points where a function becomes undefined, typically resulting from a zero in the denominator of a rational function.
Asymptotes are vital to understanding the end behavior of a graph, simplifying the analysis of complex functions, and making predictions about a graph's trends.

Horizontal Asymptotes

For the function given in the exercise, \[ g(x)=\frac{8}{1+e^{-0.5/x}} \], horizontal asymptotes offer insights into how the function behaves when the x-values become very large or very small. Horizontal asymptotes are identified by observing the function as it approaches infinity or negative infinity along the x-axis.

To determine horizontal asymptotes, we examine the limit of the function as \( x \to \infty \) or \( x \to -\infty \). For this exercise, you can notice how \( g(x) \) reaches a stable value when x becomes very large in either direction:

• As \( x \to \infty \), the term \( e^{-0.5/x} \) approaches 1, simplifying the expression to suggest \( g(x) \to 8 \).
• As \( x \to -\infty \), the function behaves similarly.Therefore, the horizontal asymptote for this function is the line \( y = 8 \).

Horizontal asymptotes don't always dictate the behavior across all parts of the graph, but they give us crucial information about the global trend towards the extremes of the x-axis.

Vertical Asymptotes

Vertical asymptotes occur at the x-values where a function is undefined. These occur when a rational function's denominator equals zero. Once identified, they can be crucial as they often represent boundaries or restrictions within a graph.

For the function in the exercise, \( g(x)=\frac{8}{1+e^{-0.5/x}} \), let's explore potential values that could cause division by zero. Unlike typical rational functions, this equation involves an exponential, \( e^{-0.5/x} \), which is always positive for all real x and never zero. Consequently, the denominator \( 1+e^{-0.5/x} \) will never equal zero for any real x-value.

Thus, this function has no vertical asymptotes as no x-value can make the denominator equal to zero, ensuring that the function remains defined across all x-values.