Question

Could someone tell me if my answers look correct?
Define a function f: R-->R by the rule f(x) = x^3-x.
(a) Is this function one-to-one? Why or why not?
(b) Is this function onto? Why or why not?
(c) Is this function invertible? Why or why not?
My answers:
(a) No b/c x could be 0, 1 or -1. I think I need to show a formal proof for this but I don't know how to write it. Any help is appreciated!
(b) No, b/c if we let y be a real number and set y=x^3-x then y does not = x...same thing...I don't know how to write a proof for this.
(c) No, b/c the function is not one to one it's not inver

Answer 1

a) correct, you can say

f(0) = 0 and f(-1) = 0 and f(1) = 0

those three inputs (x = 0, x = -1, x = 1) all produce the same output y = 0. So this counterexample shows f(x) is not one-to-one

Answer 2

of course you'd show the work on how to get f(0) = 0, f(-1) = 0, etc

Answer 3

b) all cubic functions are onto because the end behavior has both endpoints going off to +infinity and -infinity. Therefore, the range is (-infinity, infinity). So it is possible to generate any y value for some x value(s). The proof involves you finding the inverse of y = x^3 - x, but I forget how to do that with cubics.

Graph of some cubic

Answer 4

and yeah, with part c), if you don't have a one-to-one function then you can't invert it. If you need to invert it, then you have to restrict the domain to get a one-to-one region of the function.

Answer 5

Thank you so much!!