Question

Use a graphing utility to graph the function on the interval \([-4,4] .\) Does the graph of the function appear continuous on this interval? Is the function continuous on [-4,4]\(?\) Write a short paragraph about the importance of examining a function analytically as well as graphically. $$ f(x)=\frac{e^{-x}+1}{e^{x}-1} $$

Short answer

The graph of the function appears to be continuous over the interval [-4,4] which is backed by the analytical solution. Yet, despite its apparent continuity in the graphical representation, it's important to analyze the function analytically to find points of discontinuity not easily seen in the graph.

Step-by-step solution

Graph the function

Using a graphing utility, plot the function \(f(x)=\frac{e^{-x}+1}{e^{x}-1}\) on the interval [-4, 4].

Determine graphical continuity

Inspect the graph. If the graph is unbroken, then the function appears to be continuous over the interval. However, the continuity can only be fully determined analytically.

Analyze function analytically

To confirm continuity, we need to verify its limit from both sides at every point within the interval. However, since the function is a combination of exponential functions which are continuous everywhere, the given function is also continuous except where the denominator becomes zero. In the original function, checking for denominator not equal to zero gives \(e^x \neq 1\) or \(x \neq 0\). This indicates that there is a point discontinuity at \(x=0\). But since \(x=0\) does not belong to the interval [-4, 4], the function is indeed continuous on that interval.

Reflection on analytical and graphical methods

While the graphical method gives a good first impression if a function is continuous in a certain range, the analytical method is important for precise and accurate determination, especially for complex functions which cannot be easily visualized. Both methods are complementary to each other in function analysis.

Key concepts

Function Continuity

Understanding function continuity is pivotal in calculus as it ensures smooth transitions between points on a graph. A function is said to be continuous on an interval if you can draw it without lifting your pen from the paper. For our function
\(f(x)=\frac{e^{-x}+1}{e^{x}-1}\)
, we look for any potential breaks in the graph.

Nonetheless, visual inspection is not enough; we need to delve into the function's behavior analytically. This involves checking if the function is defined at each point and if the limits match the function's values from both sides. While our graph suggests continuity on the interval [-4,4], we must consider that the function has a point of discontinuity at
\(x=0\)
, which is outside the given interval. This reflects the intricate nature of continuity: a function can shift from being continuous to discontinuous with a slight change in its domain.

Graphing Utility

A graphing utility is an invaluable tool for visualizing functions. When we use these tools to plot
\(f(x)=\frac{e^{-x}+1}{e^{x}-1}\)
on the interval [-4,4], we get a clear picture of its behavior. However, rely on them with caution. For instance, they may not pick up on subtle points of discontinuity or correctly show end-behavior.

Using a graphing utility does provide immediate visual feedback and can greatly aid in understanding the overall shape and progression of a function. It's also excellent for conjecturing the behavior of functions before performing rigorous analytical computations, which makes these utilities a good starting point in the function analysis process.

Analytical Methods in Calculus

In contrast to graphical methods, analytical methods in calculus provide a mathematical framework to thoroughly understand function properties. For continuity, we employ limits. With our function, we must ensure that the limits as
\(x\)
approaches any point within our interval are equal to the function's value at that point. Seeing that exponential functions are inherently continuous and our denominator
\(e^{x}-1\)
must not be zero, we identify that except for where
\(x=0\)
, the function will remain continuous. Analytical methods also help to find exact values, intercepts, and turning points, which are necessary for complete comprehension of a function's characteristics—something that may not be readily evident in a graph alone.

Exponential Functions

Exponential functions, such as
\(e^{-x}\)
and
\(e^{x}\)
found in our given function, are fascinating due to their continuous nature over the entire real number line. This essence of exponential functions assures that any singularities or discontinuities in functions involving them are mostly due to other factors, like undefined points in rational functions.

In practice, exponential functions show up in various contexts, including population growth, radioactive decay, and investment growth, because of their consistent increase or decrease at a proportionate rate. Their unique properties allow for the construction of models that provide accurate descriptions of real-world phenomena across disciplines.