1
vote
0answers
22 views

Characterizing a function regarding symmetry

Let us suppose a function $f \colon \mathbb{N} \times \mathbb{N} \to \mathbb{R}$, such that $$\neg\left(\forall a,b \,|\, a \in N \land b \in N \implies f(a,b)=f(b,a)\right)$$ That is $$\left(\exists ...
0
votes
0answers
14 views

Is the autocorrelation of a function the same if one term is flipped on the y axis?

I have some questions about autocorrelation. They are very related, so I thought that one single post was appropriate for the topic. The first question is already illustrated in the subject: if I ...
0
votes
2answers
27 views

Create a new function using symmetry w.r.t a point

Let's say I have a function $f$ which is defined on an interval $[0,1]$ . How can I create a function $g$ defined on $[0,2]$ where: $f(x)=\sqrt{x^{a} - x^b}$ ,$0<a<b$ ,$a$ and $b$ are constants ...
0
votes
3answers
361 views

center of symmetry formula

How to prove that $I(0,-1)$ is the center of symmetry of the function $$F(x)= x - \dfrac{2e^x}{(e^x -1)}$$ Is there any formula that I can directly apply?
3
votes
1answer
33 views

Symmetry and singularities

The function $f(x) = x$ is point symmetric. But what's with $g(x) = (x^2 - x)/(x - 1)$? $\mathbb{D}_g = \mathbb{R} \backslash \{1\}$ $g(x) = -g(-x)$ is true for every $x \in \mathbb{R} \backslash ...
3
votes
3answers
912 views

Exponents in Odd and Even Functions

I was hoping someone could show or explain why it is that a function of the form $f(x) = ax^d + bx^e + cx^g+\cdots $ going on for some arbitrary length will be an odd function assuming $d, e, g$ and ...