Calculus Query

[jdiwnab]

A one-to-one function is a function that has only one value of y (f(x)) for every value of x. You could also say that it passes the "horazantal line test" where if you were to draw horizantal lines, each would only pass though the graph once. One-to-one functions are important becuase they are functions that, when inversed, are still functions. y=x^2 is a function, but not one-to-one becuase f(1) and f(-1) have the same value. y=x^3 is one-to-one becuase no values of y repeat themselves. You can inverse y=x^3 and still have a function.
y=x^3 becomes x=y^3 when inversed.
x^(1/3)=y thus is also a function.
"One-to-one" comes from the fact that there is one value of x for every value y and one value y for every value x.
Nothing says that a functions has to be one-to-one. To inverse it and still have a function, yes. But a functions has to pass the virtical line test: if virtical lines were drawn, they would pass though the graph no more than once.
y^2=x passes the horizantal line test, so you can inverse it and have a function, but doesn't pass the virtical line test so it, in itself, is not a function.
The opposite is true of y=x^2. Is a function, but can't have the inverse be a function. Infact, it is the inverse of the above, but it isn't a function on the traditional x-y axis.

Things have been drifting, but I felt the need to clear the air about what is a function and a one-to-one function.