Finding if the equation is even or odd

Question

I am learning fourier transform and I came across this question in which author right away says the given equation is "even". How does this equation become "even"?

$$x[n]=\begin{cases}A & -M\le n\le M\\0 & \text{elsewhere}\end{cases}$$

Regards

Answer 1

A function $x(n)$ is even if $x(n) = x(-n)$.

For your function there are two cases:

Case 1: $|n| \leq M$. Then $x(n) = x(-n) = A.$

Case 2: $|n| > M$, and $x(n)=x(-n) = 0.$

In both cases, $x(n) = x(-n)$, so $x$ is even.

EDIT: Incidentally, the words "even" and "odd" come from the fact that if $x(n)$ is analytic, all of the terms in its Taylor series have even powers of $n$ if $x$ is even, or odd powers of $n$ if $x$ is odd. But functions can be even or odd even if they're not differentiable, such as in your case.

Answer 2

For $-M\leq n\leq M$, $x(-n)=A=x(n)$. Elsewhere, $x(-n)=0=x(n)$.