Consider the problem with a split function f(x)= 1-x when x<1 and sqrt(x-1) when x>or=1

Why is that not considered 1 to 1 ? In a chapter of a calculus book before they get into limits, I can see that neither side of 1 has a limit from both directions. But the limit of f(x) from the left of x<1 , (1-x)=0 and the limit of f(x) from the right of x>=1 in sqrt(x-1)=0

Also since the union of the range of (1-x) and sqrt(x-1) is (-inf,1) U [1,inf) = (-inf,inf) every value of x is mapped to a single value of f(x) so whats the problem ? So why's that not considered 1:1 ? What am I forgetting or missing from pre-calculus math that says that's not 1:1

 

I'd guess it is because f(0) = f(2) etc. Been decades since I studied this, but IIRC f(x)==f(y) implies x==y for a 1 to 1 function.

 

So only functions that map each x to a unique f(x) value.
Okay so any f(x) whose slope / derivative crosses though 0 would also not be considered 1:1 then.
I've done all this in Discrete math class before too come to think of it. I guess I should start reviewing them too.

 

Consider f(x)=x*x is not 1:1 but f(x)=x*x*x is 1:1.

 

yup