Questions tagged [even-and-odd-functions]

Even functions have reflective symmetry across the $y$-axis; odd functions have rotational symmetry about the origin.

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What does $f(x)f(-x) \le 0$ mean when $f(x)$ is a cubic function?

Like the title, what does $\forall x\in \mathbb{R}, f(x)f(-x) \le 0$ say about $f(x)$, when $f(x)$ is a cubic function? The book says $f(x)$ has to be the odd function, but I can't figure out rigorous ...
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Proving that $y = a_{2n}x^{2n} + a_{2n−2}x^{2n−2} + . . . + a_2x^2 + a_0$ is an even function.

This question is from Larson's Precalculus book. Prove that a function of the form $$y = a_{2n}x^{2n} + a_{2n−2}x^{2n−2} + . . . + a_2x^2 + a_0$$ is an even function. I don't understand how that ...
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The sine transform of symmetric functions

I am trying to think about the sine transform of symmetric functions, chiefly the unit rectangle and the unit triangle function. I am poor with LaTeX so I'll do this as best I can Say I have the ...
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Finding Fourier series of $f(x)+c$ given that of $f(x)$

So, I have the function $f(x)$ over the interval $[-\pi,\pi]$ defined as under. $$f(x)=\begin{cases}1+2x/\pi , -\pi\le x\le 0 \\ 1-2x/\pi , 0< x\le \pi\end{cases}$$ The thing is computing the ...
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Determining if $f(x) = \frac{x}{x+1}$ is Even or Odd

This is likely a silly question, but I cannot seem to wrap my head around the following: I am trying to determine whether the following function is even or odd: $f(x) = \frac{x}{x+1}$. I pass $-x$ as ...
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Error in solution of 3-45 in Solution Manual of Signals and Systems Oppenheim 2nd edition

Hello I was solving the question 3-45 of the book Signals and Systems 2nd edition and I assume there is an error in the solution manual, kindly if someone confirm that am I right? or the solution ...
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Find $f(x)$ from two-variable equation

Let $f$ be a function which satisfies the equation \begin{equation} f(x-y+a)-f(x+y+a)=2f(x)f(y) \end{equation} where $a$ is a positive real number and for every pair of real numbers $x$ and $y$. ...
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Even, odd, or neither $y=\frac{1-\cos{x}}{1+\cos{x}}$

Determine algebraically whether the given function is even, odd, or neither. $$y=\dfrac{1-\cos{x}}{1+\cos{x}}$$ I will substitute $−x$ into the function, and then simplify. $$y(-x)=\dfrac{1-\cos(-x)}{...
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Which even function $f(x)$ satisfies $\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\cos(nx)~dx=\frac{1}{n}$?

Question: Which even function $f(x)$ satisfies $$\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\cos(nx)~dx=\frac{1}{n}? $$ Alternatively, which even function $f(x)$ satisfies $$\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\...
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Integrate $\int^\infty_0 \dfrac{\sin x}{x^3}\,dx$

By using this method we can evaluate $\displaystyle\int^\infty_0\dfrac{x-\sin x}{x^3}\,dz=\dfrac{\pi}{4}$ and I intended to solve $\displaystyle\int^\infty_0 \dfrac{\sin x}{x^3}\,dx=\displaystyle\int^\...
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Fourier transformation of a multidimensional real and even function

The fourier transformation of an one-dimensional real and even function can be simplified to $$ F(s) = 2 \cdot \int_0^{\infty} f(t) \cos(2\pi s t) \, \mathrm{d}t,\\ \text{with} \quad f(t) \in \mathcal{...
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Proving that the odd coefficients of the polynomial that interpolates an even function are zero

Let $n \in \mathbb{N}$ be fix, $a>0$ and $f:[-a,a] \rightarrow \mathbb{R}$ be an even funciton. Consider the polynomial $p(x) = \sum\limits_{i =0}^n c_ix^i \in \mathbb{P}_n$ that interpolates $f$ ...
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Integrals, and even and odd functions

I am looking at a proof and got stuck on a part with an integral. I tried to simplify the problem as much as possible, I hope I did not omit any potential helpful information. I have an even ...
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Is $\cos(a(\pi-t))$ an even function?

I've an assignment due in a couple of days, and I'm wondering if my teacher made a mistake in the question below, or if I'm missing something silly. Let $a \in [0, 1]$ and $f_a$ the $2\pi$-periodic ...
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Odd function with no critical points must be linear.

Let $f: \mathbb{R}\to \mathbb{R}$ be a differentiable odd function, i.e., $f(x) + f(-x) = 0$ for all $x\in \mathbb{R}$ and $f^\prime(x)$ exists for all real $x$. Further assume the followings: $ \...
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How to transition from approximation of odd function using $\sin$ and $\cos$ to only using $\sin$?

I'm failing to understand the following transition: Given that $f$ is an odd function defined on $[-\pi,\pi]$, we know that for all $\epsilon>0$ there exist $a_n, b_n \in \mathbb{R}$ such that: $$||...
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Obtain a function by it's even or odd part

Since every function can be divided in a even and a odd part: \begin{equation} f_e(x)=\frac{f(x)+f(-x)}{2} \end{equation} \begin{equation} f_o(x)=\frac{f(x)-f(-x)}{2} \end{equation} \begin{equation} f(...
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What are these 'partial' reflective invariants and equivariants of a multiary function called?

In the univariable case we say that a function is even if $f(x)=f(-x)$ and odd if $-f(x)=f(-x)$ for all $x$ in some space of interest. In the multiary case we would similarly consider a function to be ...
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Derivation of function$~g(x)=f(x)+f(-x)~$is an even function where$~f(x)~$is an arbitrary continuous function.

Let$~f~$be any function which is defined for all numbers. Show that$~g(x)=f(x)+f(-x)~$is even. $$\operatorname{e.g.}~f(x)=x^2\implies g(x)=x^2+(-x)^2=2x^2\leftarrow~~\text{even}\tag{1}$$ $$\...
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prove or disprove: Multiplication of infinite number of even functions is also an even function

I know that if $f_1(x)$,$f_2(x)$,...,$f_n(x)$ are even functions and $c_1$,$c_2,...,c_n$ are fixed real numbers then $c_1f_1(x)$ $c_2f_2(x)$ ... $c_nf_n(x)$ is also an even function. But how for ...
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prove or disprove that "the sum of infinite even or odd functions is even or odd".

i know that if $f_1(x),f_2(x),...,f_n(x)$ is odd/even functions and $c_1,c_2,...,c_n$ are fixed real numbers then $c_1 f_1(x) + c_2 f_2(x) +... +c_n f_n(x)$ is odd/even. But how for infinite sum ((sum ...
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Construct a function $f$ with $f'-af$ is odd (a>0)

Let $a>0$, I am trying to construct a function $f$ such that $f'-af$ is odd. i.e \begin{align*} f'(-x)-af(-x)=-f'(x)+af(x) \end{align*} By direct computation, we have \begin{align*} \frac{d}{dx}(f(...
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It makes sense to consider oddness with respect a point which is not $0$?

Let $F$ be a $C^1$ function and consider $$G(x) =ax- F(x+a),$$ with $a\in\mathbb{R}^*_+$. I need $G$ to be even. Clearly, if $F$ is odd in the "usual" sense, so it is $G$. But, actually, I ...
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solving definite integral with properties

Could someone help me with this question? Suppose that f(x) is a continuous and odd function such that $$\displaystyle\int^7_2f(x)\ d(x)=-3$$ Find $$\int^7_{-2}(1+f(x))\ dx$$ So far, I have used the ...
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1 answer
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What is the decomposition into even and odd function of a function equal to $x$ when $x \geq 0$ and equal to zero when $x <0$?

Every function $f$ with domain in $\mathbb{R}$ can be written $$f=E+O$$ where $E$ is an even function and $O$ is an odd function. Proof Assume $f(x) = E(x) + O(x)$. Then $$f(-x)=E(-x) + O(-x)=E(x)-O(x)...
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Constructing holomorphic odd functions

Suppose $f(z)$ is holomorphic on $\Re(z) \geq 0$ satisfying $f(-it) = -f(it)$ for any $t \in \mathbb{R}$. Can $f$ be analytically continued to an entire function $g$ on $\mathbb{C}$ satisfying $g(-z) =...
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Can I see from here if this integral is zero?

I'm computing the following integral: $$\int_0^{\pi/4}\int_0^{2\pi} \sin^2(\rho)\sin(\theta)-\cos(\rho)\sin^2(\rho)\sin(\theta)~d\theta d\rho$$ Till this point I did everything correct, since when I ...
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1 answer
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Solving summation of fractional powers with floor

$\sum_{i=0}^{k-1} 2^{\lfloor{i/2}\rfloor} = 2 * (2^{k/2}-1)$ when k is even $\sum_{i=0}^{k-1} 2^{\lfloor{i/2}\rfloor} = 3 * 2^{(k-1)/2}-2$ when k is odd How can I solve this by induction or derive the ...
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Sum of roots for a even function and slope being 0?

High school senior and was helping my friend with Rolle's theorem and came across this neat point. Given $f(x)$ is a continuous and differentiable function, and $f(-x)=f(x)$. Additionally, $f(x)=0$ ...
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Would this be an even or odd function?

Given that $f(x)$ is odd, I need to find if $f(\sec x)$ is odd. We (or I, at least) have always defined an odd function in the following way: $f(x)$ is odd if $f(x)=-f(-x)$, for all $x$ in the domain ...
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Is this function odd with respect to $x$?

If $g(x)$ is odd with respect to $x$, then is $$\int_{x-ct}^{x+ct}g(s)ds$$ also odd with respect to $x$? I tried by fundamental theorem of calculus that $$-\int_{x-ct}^{x+ct}g(s)ds=-(G(x+ct)-G(x-ct))$$...
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For any $(a,b)\in\mathbb R^2$ such that $2f(a)+f(-b)\geq 0$, we have: $2f(-a)+f(b)\leq 0$. Then, $f$ must be an odd function?

For any $(a,b)\in\mathbb R^2$ such that $2f(a)+f(-b)=0$, we have: $2f(-a)+f(b)=0$. For any $(a,b)\in\mathbb R^2$ such that $2f(a)+f(-b)>0$, we have: $2f(-a)+f(b)<0$. Also, for all $a$, there ...
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$f(a)+f(-b)=0$ implies $f(-a)+f(b)=0$. Then $f$ is an odd function?

If $f(a)+f(-b)=0$ for some $(a,b)$, then, $f(-a)+f(b)=0$. Also, for all $a$, there exists $b$ such that $f(a)+f(-b)=0$. $f:\mathbb R\to\mathbb R$ is continuous. Based on the above conditions, can we ...
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Even Extension of Heat Equation on Semi-Infinte Domain

I am working on the heat equation in a semi-inf domain. I am doing an even extension: $$u(x,t)= \int_{-\infty}^0 G(x-y,t)f(-y)dy + \int_{0}^\infty G(x-y,t)f(y)dy $$ Now in the next step they do: $$u(x,...
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Why is $\hat f(\xi)/\xi\in L^1(\mathbb R)$ when $f\in L^1$ is odd,$\hat f'(0)$ exists, and $\hat f(\xi)$ always has the same sign as $\xi$?

I need to prove: If $f \in L^1(\mathbb{R})$ is an odd function such that its Fourier Transform $\hat{f}$ is differentiable at $\xi = 0$ and $\hat{f} \geq 0$ when $\xi \geq 0$, then $\hat{f}(\xi)/\xi \...
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4 answers
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Is $f(\theta) = 1 - \alpha \sin{\theta}$ an odd function?

Consider equations in polar coordinates of the form $$r = f(\theta) = 1 - \alpha \sin{\theta}$$ When I plot a few of these polar functions, I always get a graph that is symmetric relative to the $y$ ...
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Do holes affect the type of function (even, odd or neither)

We can know whether a function is even or odd by substituting using F(-x) but what if the function has a single hole like this f(x) = $\frac{x(x-2)}{x-2}$ is such a function considered odd or neither? ...
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Prove certain integral equals $0$

I've been trying to solve some problems from my Partial Differential Equations course. I got the idea to solve this one but there's one step at the end I'm not sure about. It goes like this: Given $$...
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Symmetry in Fourier and inverse Fourier transform

Fourier transform and inverse Fourier transform is given as - $f(x)=\int_{-\infty}^{\infty}g(\alpha)e^{i\alpha x}d\alpha$ $g(\alpha)=\frac{1}{2\pi}\int_{-\infty}^{\infty}f(x)e^{-i\alpha x}dx$ I am ...
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For a spherically symmetric Riemannian manifold, are all the metric functions even or odd?

For a spherically symmetric Riemannian manifold, we can write the metric as $ds^2 = a^2(r)dr^2 + b^2(r)d\Omega^2$ Now I think the metric components have to be even functions for the manifold to be ...
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Classifying the function as even or odd in restricted domain

Let $f:(-\pi,\pi)\rightarrow[0,1], f(x)=\cos(x)$ Clearly this function is even as $f(x)=f(-x)$ for all $x\in (-\pi,\pi)$ Suppose if we defined $f$ as $f:[0,\pi)\rightarrow[0,1], f(x)=\cos(x)$ How do ...
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Necessary condition for a function to be odd

Let $f:\mathbb R\to\mathbb R$ such that it holds $$\int_{-a}^af(x)dx=0\qquad \forall a\in\mathbb R^+.$$ Is it true then that the function $f$ is odd, i.e. $f(x)=-f(-x)?$
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D'Alembert functional equation without odditivity

Problem Find the functional $f: \mathbb{R} \to \mathbb{R}$ with $f(x+y)+f(x-y)=2f(x)cosy$ Attempts Now as I know, the functional equation can be solved more generally as follow: General problem Find ...
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Why does WolframAlpha give me an ugly graph for the input "(cos x-1)/x^2 when x=0.01"?

The function $\dfrac{\cos x-1}{x^2}$ is differentiable everywhere except for $x=0$. However, when I give WolframAlpha the input "(cos x-1)/x^2 when x=0.01", it gives me an ugly graph as ...
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Even And Odd Analogous To Plus And Minus

How is even and odd related to + and - because 2 odd functions or 2 even functions (and permutation groups too) are multiplied together to give an even result. If they are different like odd even then ...
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2 answers
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Is $\ln(\sqrt{x^2 + 1} - x)$ an odd function?

$f(x)$ is an odd function if $f(-x) = -f(x)$. If $f(x) = \ln(\sqrt{x^2 + 1} - x)$, one can observe graphically that $f(-x) = -f(x)$. Hence, $f(x)$ must be an odd function. However, WolframAlpha gives ...
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3 votes
2 answers
144 views

What does it mean for a function to be oddly and evenly symmetric?

From understanding, a function describes a relationship between multiple variables, and has unique values across all possible values on one axis while not duplicating violating a vertical line test to ...
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A function satisfying $f(x+y)+f(x-y)=2f(x)f(y)$ and $f(x_0)=-1$ is periodic

Suppose that $f:\mathbb{R}\to \mathbb{R}$ satisfying that $f(x+y)+f(x-y)=2f(x)f(y)$ forall $x,y\in \mathbb{R}$ and there exists $x_0$ such that $f(x_0)=-1$. Prove that $f$ is a periodic function. ...
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How do we expand non odd functions in Fourier sine integral transform

Find the Fourier sine integal for $f(x)= e^{-bx}$ and prove that $$\frac{\pi}{2}f(x)=\int_0^{\infty}\frac{t\sin(tx)}{b^2+t^2}dt.$$ I don't want the solution to the problem, I was able to solve it. ...
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Showing if the following function is odd or even

The function $f$ is periodic with $T=2\pi$ and is given as $f(t)=t$ where $-\pi \leq t \leq \pi$ Is $f$ odd, even or periodic or neither? Also, where is $f$ discontinuous? Normally if a function is ...
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