# Questions tagged [even-and-odd-functions]

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

245 questions
Filter by
Sorted by
Tagged with
59 views

### 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 ...
50 views

### 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 ...
1 vote
31 views

### 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 ...
56 views

### 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 ...
54 views

### 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 ...
31 views

### 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 ...
36 views

### 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 $y$. ...
51 views

77 views

57 views

### 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 ...
1 vote
53 views

### 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 ...
44 views

### 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, $f(x)=0$ ...
58 views

### 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 ...
47 views

### 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 $$-\int_{x-ct}^{x+ct}g(s)ds=-(G(x+ct)-G(x-ct))$$...
44 views

### 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 ...
69 views

### $f(a)+f(-b)=0$ 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 ...
14 views

67 views

### 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 ...
43 views

### 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 ...
31 views

### 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 ...
1 vote
54 views

### 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. $f(x)=-f(-x)?$
81 views

### 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 ...
57 views

### 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 ...
54 views

### 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 ...
112 views

### 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 ... 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 ...
109 views

### 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. ...
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. ...
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 ...