Question

How to determine which of the following functions are one-to-one ?

Answer 1

See explanation...

Explanation:

A. `f` is not one-one. The function maps all the points of `RR^3` to the plane `x+y+z = 0`.

For example:

`f(1, 1, 1) = (1-1, 1-1, 1-1) = (0, 0, 0) = (2-2, 2-2, 2-2) = f(2, 2, 2)`

B. `f` is not one-one since it is an even function `f(-1) = 1 = f(1)`

C. `f` is one-one (essentially a rotation and scaling) with inverse `f^(-1)(x, y, z) = ((x-y+z)/2, (y-z+x)/2, (z-x+y)/2)`

D. `f` is one-one (a rotation and scaling) with inverse `f^(-1)(x, y) = ((x+y)/2, (x-y)/2)`

E. `f` is not one-one `f(1) = 1 - 1 = 0 - 0 = f(0)`