Changes

The function <math>f(x) = x^2</math> fails to be onto because there is no input that gives a result of, say, -3.  It fails to be one-to-one because <math>f(3)</math> and <math>f(-3)</math> are the same number.

The function <math>f(x) = x^2</math> fails to be onto because there is no input that gives a result of, say, -3.  It fails to be one-to-one because <math>f(3)</math> and <math>f(-3)</math> are the same number.

The fact that a function is a one-to-one correspondence does not mean that its domain and range are the same.  There is, for example, a one-to-one correspondence between the set of all reals and the interval <math>(-\pi/2, \pi/2)</math>.  The arctangent function does it.

The fact that a function is a one-to-one correspondence does not mean that its domain and image are the same.  There is, for example, a one-to-one correspondence between the set of all reals and the interval <math>(-\pi/2, \pi/2)</math>.  The arctangent function does it.

==Composition==

==Composition==