Which functions are one-to-one? Which functions are onto? Describe the inverse function

A)F:Z^2-N where f is f(x,y) x^2 +2y^2
B)F:N->N where f is f(x) = x/2 (x even) x+1 (x odd)
C)F:N->N where f is f(x) = x+1 (x even) x-1 (x odd)
D)h:N^3 -> N where h(x,y,z) = x + y -z

Solution Preview

<br>A) f is not one-to-one since f(-1,1)=f(1,1)=3. f is not onto since f(x,y)=x^2+2y^2=2 has no integer solutions. There is no inverse function.
<br>B) f is onto since for each n in N, we have f(2n)=2n/2=n. f is not one-to-one since f(4)=4/2=2 and ...

Solution Summary

This shows how to identify inverse, one-to-one, and onto functions.

... c) Define a function h: X Y that is neither onto nor one-to-one. ... c) Define a function h: X Y that is neither onto nor one-to-one. Solution: ...

... impossible. b) f(x) = 10+x is both one to one and onto c) f(x) = 10+x is not onto in this case. ... b) f(x) = 10+x is both one to one and onto. c ...

Consider an arbitrary mapping f : X -->Y. Suppose that f is a one-to-one onto. ... Consider an arbitrary mapping f : X -->Y. Suppose that f is a one-to-one onto. ...