Question

Define the following with two examples
Zeros function
Even function
Odd function
Heaviside function
Greatest integer function
Absolute function
Identity function
Constant function
Periodic function

Answer 1

1) Zero function: This is a function equal to zero on every point of its domain ($y=0, y=0(x+1)$ );

2) Even function: For every x of this type of function $f(-x)=f(x)$ ($y=x^2, y=cos(x))$

3) Odd function: For every x of this type of function $f(-x)=-f(x)$ ($y=x^3, y=sin(x)$)

4) Heaviside function: This is a unit step function $H(x):=\frac{d}{dx}max(x,0)$ ($H(x)=\frac{1}{2}+\frac{1}{2}sgn(x), H(x)=1_{[0, infinity)}(x)$)

5) Greatest integer function: The Greatest integer function is defined as the largest integer which is less than or equal to x ($y=[x], y=\frac{1}{2}[x]$)

6) Absolute number: This is a positive number for positive x and the opposite number for a negative x ($y=\mid{x}\mid, y=\mid{x+1}\mid)$

7) Identity function: This is actually the same value that was used as an argument ($y=x$)

8) Constant function: This is the function with the same value for every argument x ($y=3, y=3.14$)

9) Periodic function: For every x of this function $f(x)=f(x+a)$, where a is a non-zero real number or period ($y=sin(x), y=cos(x)$)