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Chapter 5: Problem 22

Determine whether the function is even, odd, or neither . $$f(x)=\frac{e^{x}-e^{-x}}{2}$$

Short Answer

Expert verified
Answer: The function is odd.

Step by step solution

01

Determine f(-x)

First, we need to replace \(x\) with \(-x\) in the given function: $$f(-x)=\frac{e^{-x}-e^{x}}{2}$$
02

Evaluate if f(-x)-f(x)=0 or f(x)+f(-x)=0

Now, we need to determine if \(f(-x) = f(x)\) for even functions or \(f(-x) = -f(x)\) for odd functions: To check if the given function is even, we should prove that \(f(-x) - f(x) = 0\): $$\frac{e^{-x}-e^{x}}{2} - \frac{e^{x}-e^{-x}}{2} = \frac{e^{-x}-e^{x}-(e^{x}-e^{-x})}{2}= \frac{e^{-x}-e^{x}-e^{x}+e^{-x}}{2}= \frac{2e^{-x}-2e^{x}}{2}=\not 0$$ So the function is not even. Now, let's check if the given function is odd by testing if \(f(x) + f(-x) = 0\): $$\frac{e^{x}-e^{-x}}{2} + \frac{e^{-x}-e^{x}}{2} = \frac{e^{x}-e^{-x}+e^{-x}-e^{x}}{2} = \frac{2e^{-x}-2e^{x}+2e^{-x}+2e^{x}}{2}=0$$ Since \(f(x) + f(-x) = 0\), the function is odd.
03

Conclusion

The function $$f(x)=\frac{e^{x}-e^{-x}}{2}$$ is an odd function.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Even Functions
An even function is a function that remains unchanged when you replace its input variable with the opposite sign. In simpler terms, if you have a function \(f(x)\), it is classified as even if for every value of \(x\), the equation \(f(-x) = f(x)\) holds true. This means the graph of an even function is symmetric with respect to the y-axis.
To remember the characteristics of even functions, think about classical symmetric shapes, just like circles. All even functions reflect this type of balance. Some classic examples include polynomial functions with only even powers, such as \(f(x) = x^2\) or \(f(x) = x^4\).
  • Key characteristic: Symmetric about the y-axis.
  • Formula: For all \(x\), \(f(-x) = f(x)\).
  • Common examples: Cosine function \(\cos(x)\), quadratic functions like \(x^2\).
Function Symmetry
Understanding function symmetry is crucial in classifying functions as odd, even, or neither. Symmetry can make functions easier to analyze, especially when calculating integrals or understanding their behavior across axes.
There are two main types of function symmetry:
  • Y-axis symmetry: Even functions possess this property. They show identical behavior on both sides of the y-axis, meaning the graph behaves the same if you fold it over this axis.
  • Origin symmetry: This is a hallmark of odd functions. Such graphs look the same after rotating 180 degrees around the origin. This symmetry means \(f(-x) = -f(x)\).
Identifying symmetry in a function assists in predicting its values without explicit calculations, particularly helpful for visualizing graphs and solving equations. For odd functions like the one in our problem \(f(x)=\frac{e^{x}-e^{-x}}{2}\), the presence of origin symmetry confirms that it is indeed an odd function.
Exponential Functions
Exponential functions are a fundamental part of mathematics and involve expressions with a constant base raised to a variable exponent. A general form of an exponential function can be seen as \(f(x) = a^x\), where \(a\) is a positive constant.
These functions exhibit rapid growth or decay. When the base is greater than one, as \(x\) increases, \(f(x)\) grows fast. Conversely, if the base is between zero and one, the function decays as \(x\) increases.
In the context of the function \(f(x) = \frac{e^x - e^{-x}}{2}\):
  • \(e^x\) represents exponential growth, a key component in calculus, often linked with continuous compounding.
  • \(e^{-x}\) signifies exponential decay, illustrating processes that slow down over time.
  • The combination \(e^x - e^{-x}\) creates a hyperbolic sine function, which exhibits odd symmetry, as seen by checking \(f(-x) = -f(x)\).
Exponential functions are also pivotal in describing real-world phenomena, such as population growth, radioactive decay, and interest calculations.

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Most popular questions from this chapter

Deal with the compound interest formula \(A=P(1+r)^{t},\) which was discussed in Special Topics \(5.2.A\). At what rate of interest (compounded annually) should you invest 500 dollars if you want to have 1500 dollars in 12 years?

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Deal with functions of the form \(f(x)=P e^{k x}\) where \(k\) is the continuous exponential growth rate (see Example 6 ). The present concentration of carbon dioxide in the atmosphere is 364 parts per million (ppm) and is increasing exponentially at a continuous yearly rate of \(.4 \%\) (that is, \(k=.004) .\) How many years will it take for the concentration to reach 500 ppm?

In the past two decades, more women than men have been entering college. The table shows the percentage of male first-year college students in selected years. $$\begin{array}{|l|l|l|l|l|l|l|l|l|l|} \hline \text { Year } & 1985 & 1990 & 1995 & 1997 & 1998 & 1999 & 2003 & 2004 & 2005 \\ \hline \text { Percent } & 48.9 & 46.9 & 45.6 & 45.5 & 45.5 & 45.3 & 45.1 & 44.9 & 45.0 \\ \hline \end{array}$$ (a) Find three models for this data: exponential, logarithmic, and power, with \(x=5\) corresponding to 1985 (b) For the years \(1985-2005,\) is there any significant difference among the models? (c) Assume that the models remain accurate. What year does each predict as the first year in which fewer than \(43 \%\) of first-year college students will be male? (d) We actually have some additional data: $$\begin{array}{|l|c|c|c|c|} \hline \text { Year } & 2000 & 2001 & 2002 & 2006 \\ \hline \text { Percent } & 45.2 & 44.9 & 45.0 & 45.1 \\ \hline \end{array}$$ Which model did the best job of predicting the new data?

The population of Mexico was 100.4 million in 2000 and is expected to grow at the rate of \(1.4 \%\) per year. (a) Find the rule of the function \(f\) that gives Mexico's population (in millions) in year \(x,\) with \(x=0\) corresponding to 2000. (b) Estimate Mexico's population in 2010 . (c) When will the population reach 125 million people?
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