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Chapter 10: Problem 6

Determine whether the given function is even, odd, or neither. $$ e^{-x} $$

Short Answer

Expert verified
Answer: The function $$f(x) = e^{-x}$$ is neither even nor odd.

Step by step solution

01

Write down the given function

The given function is: $$f(x) = e^{-x}$$
02

Calculate f(-x)

Now find $$f(-x)$$ by replacing $$x$$ with $$-x$$ in the function: $$f(-x) = e^{-(-x)}$$ Simplify the exponent: $$f(-x) = e^x$$
03

Determine if f(-x) is equal to f(x) (Even Function Test)

To test if the function is even, we will check if $$f(-x)$$ is equal to $$f(x)$$. If they are equal, the function is even. Given function: $$f(x) = e^{-x}$$ Calculated function: $$f(-x) = e^x$$ Since $$f(-x) \neq f(x)$$, the function is not even.
04

Determine if f(-x) is equal to -f(x) (Odd Function Test)

To test if the function is odd, we will check if $$f(-x)$$ is equal to $$-f(x)$$. If they are equal, the function is odd. Given function: $$-f(x) = -e^{-x}$$ Calculated function: $$f(-x) = e^x$$ Since $$f(-x) \neq -f(x)$$, the function is not odd.
05

Conclusion

Since the function $$f(x) = e^{-x}$$ is neither even nor odd, we can conclude that it is neither.

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Key Concepts

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

Exponential Functions
Exponential functions are a type of mathematical function characterized by a constant base raised to a variable exponent. Typically expressed in the form \( f(x) = a^x \), where \( a \) is a constant greater than zero. These functions are frequently encountered in various fields such as physics, finance, and biology. Exponential functions have unique properties:
  • They always produce positive values and are asymptotic to the x-axis. This means the graph gets incredibly close to the x-axis but never actually touches it.
  • Exponential growth or decay, depending on whether the base is greater than or less than one, respectively.
  • They can model phenomena like population growth, radioactive decay, and interest calculations.
In our example, \( e^{-x} \), the base \( e \) (approximately 2.718) is the natural exponential base, commonly used in continuous growth models and natural processes. Replacing \( x \) with \( -x \) changes the direction of growth from increasing to decreasing, illustrating exponential decay.
Function Symmetry
Function symmetry is an important concept in mathematics that helps to classify functions into either even, odd, or neither. This classification is based on how the function behaves when the input variable, \( x \), is replaced with \( -x \).
  • Even functions are symmetrical about the y-axis. This means if you draw a vertical line at \( x = 0 \), each half of the graph should mirror the other. For these functions, \( f(x) = f(-x) \).
  • Odd functions exhibit rotational symmetry about the origin. They satisfy \( f(-x) = -f(x) \), meaning the function flips over the origin.
In our problem, we have the function \( e^{-x} \). By testing its symmetry, we found that neither \( f(-x) = e^x \) equals \( f(x) \) nor does \( f(-x) = -f(x) \). Hence, it is classified as neither even nor odd. This outcome is fairly common in functions where symmetry doesn't exist, showcasing the diverse behavior functions can possess.
Mathematical Analysis
Mathematical analysis examines the properties and behaviors of functions using logical reasoning and mathematical rigor. Part of this analysis includes determining function types, like even or odd, which can greatly influence their graphical representation and equation solving strategies.
Function behavior, such as asymptotic tendencies in exponential functions, continuity, and limits, is evaluated using various analytical tools. These include basic calculus tactics such as differentiation and integration.
Analyzing \( f(x) = e^{-x} \) involves determining that it doesn't fit the straightforward molds of even or odd functions. This understanding is crucial when attempting to find real-world applications or further manipulations of the function. These mathematical insights lay the groundwork for deeper exploration into dynamic systems or natural phenomena represented by such functions, providing essential foundational knowledge in graduate-level mathematics and beyond.

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Most popular questions from this chapter

A function \(f\) is given on an interval of length \(L .\) In each case sketch the graphs of the even and odd extensions of \(f\) of period \(2 L .\) $$ f(x)=\left\\{\begin{array}{ll}{0,} & {0 \leq x<1} \\ {x-1,} & {1 \leq x<2}\end{array}\right. $$

(a) Find the solution \(u(r, \theta)\) of Laplace's equation in the semicircular region \(r

More Specialized Fourier Scries. Let \(f\) be a function originally defined on \(0 \leq x \leq L\). In this section we have shown that it is possible to represent \(f\) either by a sine series or by a cosine series by constructing odd or even periodic extensions of \(f,\) respectively. Problems 38 through 40 concern some other more specialized Fourier series that converge to the given function \(f\) on \((0, L) .\) $$ \begin{array}{l}{\text { Let } f \text { be extended into }(L, 2 L] \text { in an arbitrary manner. Then extend the resulting function }} \\ {\text { into }(-2 L, 0) \text { as an odd function and elsewhere as a periodic function of period } 4 L \text { (see }} \\ { \text { Figure }10.4 .6) . \text { Show that this function has a Fourier sine series in terms of the functions }} \\\ {\sin (n \pi x / 2 L), n=1,2,3, \ldots . \text { that is, }}\end{array} $$ $$ f(x)=\sum_{n=1}^{\infty} b_{n} \sin (n \pi x / 2 L) $$ where $$ b_{n}=\frac{1}{L} \int_{0}^{2 L} f(x) \sin (n \pi x / 2 L) d x $$ $$ \text { This series converges to the original function on }(0, L) $$ (Figure cant copy)

In this problem we indicate certain similarities between three dimensional geometric vectors and Fourier series. (a) Let \(\mathbf{v}_{1}, \mathbf{v}_{2},\) and \(\mathbf{v}_{3}\) be a set of mutually orthogonal vectors in three dimensions and let \(\mathbf{u}\) be any three-dimensional vector. Show that $$\mathbf{u}=a_{1} \mathbf{v}_{1}+a_{2} \mathbf{v}_{2}+a_{3} \mathbf{v}_{3}$$ where $$a_{i}=\frac{\mathbf{u} \cdot \mathbf{v}_{i}}{\mathbf{v}_{i} \cdot \mathbf{v}_{i}}, \quad i=1,2,3$$ Show that \(a_{i}\) can be interpreted as the projection of \(\mathbf{u}\) in the direction of \(\mathbf{v}_{i}\) divided by the length of \(\mathbf{v}_{i}\). (b) Define the inner product \((u, v)\) by $$(u, v)=\int_{-L}^{L} u(x) v(x) d x$$ Also let $$\begin{array}{ll}{\phi_{x}(x)=\cos (n \pi x / L),} & {n=0,1,2, \ldots} \\ {\psi_{n}(x)=\sin (n \pi x / L),} & {n=1,2, \ldots}\end{array}$$ Show that Eq. ( 10 ) can be written in the form $$\left(f, \phi_{n}\right)=\frac{a_{0}}{2}\left(\phi_{0}, \phi_{n}\right)+\sum_{m=1}^{\infty} a_{m}\left(\phi_{m}, \phi_{n}\right)+\sum_{m=1}^{\infty} b_{m}\left(\psi_{m}, \phi_{m}\right)$$ (c) Use Eq. (v) and the corresponding equation for \(\left(f, \psi_{n}\right)\) together with the orthogonality relations to show that $$a_{n}=\frac{\left(f, \phi_{n}\right)}{\left(\phi_{n}, \phi_{n}\right)}, \quad n=0,1,2, \ldots ; \quad b_{n}=\frac{\left(f, \psi_{n}\right)}{\left(\psi_{n}, \psi_{n}\right)}, \quad n=1,2, \ldots$$ Note the resemblance between Eqs. (vi) and Eq. (ii). The functions \(\phi_{x}\) and \(\psi_{x}\) play a role for functions similar to that of the orthogonal vectors \(v_{1}, v_{2},\) and \(v_{3}\) in three-dimensional

Find a solution \(u(r, \theta)\) of Laplace's equation inside the circle \(r=a,\) also satisfying the boundary condition on the circle $$ u_{r}(a, \theta)=g(\theta), \quad 0 \leq \theta<2 \pi $$ Note that this is a Neumann problem, and that its solution is determined only up to an arbitrary additive constant. State a necessary condition on \(g(\theta)\) for this problem to be solvable by the method of separation of variables (see Problem 10 ).
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