There several ways to prove if a mathematical formula is a function or not :
First: To find 2 or more $X$'s that have the same $Y$ assigned to them .
Second: To assume that we put $(x_1,y_1)$ and $(x_2,y_2)$ into the formula and then come to the conclusion that $y_1=y_2$ .
But I've heard that there's even a third way which is using derivative to determine if a relation is a function or not . Honestly I cannot use the 2 provided ways efficiently in a reasonable time , when things get complicated as bellow :
Determine if bellow formulas are function or not :
- $F(X-2)=|X|-1$
- $F(|X|-1)=2X+1$
- $F(|X|-1)=2X^2 +1$
- $F(2X^2 +1)=|X|-1$
- $Y!=X$
- $Y= X + [X]$
For what amount of a , the below is representing a function ? $$(Y-X^2)( (X-2)^2 + (Y+a) ) = 0$$
Many thanks
Mohammad from Iran
