- #1

- 1

- 0

**f : N --> N defined by f(x) = x^3 - 1**

Hi all, I need answers and EXPLANATION to the following problems: (Please Help!)

(i)

*f*: N --> N defined by

*f*(x) =

*x*^3 - 1

(ii)

*g*: Z --> Z defined by

*g*(x) = 2

*x*+ 1

(iii)

*h*: R --> R defined by

*h*(x) =

*x*(

*x*+ 3)(

*x*- 3)

(***note that N,Z,R stands for natural #, Integer, and Real # respectively..)

(a) Which of the functions are one-to-one?

(b) Which of the functions are bijections?

(c) For those that are bijections find the inverse function.

--------------------------------------------------------------------------------------------------------------

Here's the other one:

The function

*f*: R --> R defined by

*f*(x) = (3

*x*- 1)/(

*x*- 3) is not bijective however by suitably restricting the domain and codomain the function can be made to be bijective.

(a)State the domain and codomain that will make the function bijective.

What's a domain? codomain?

(b) Find the inverse of the bijective function.

(I can still remember a bit of inverse function.. i think.. well ill give it a try anyway)

*f*(x) = (3

*x*- 1)/

*x*- 3)

*x*= (3

*y*- 1)/

*y*- 3) "replace x with y"

*x*(

*y*- 3) = 1(3

*y*- 1)) "Cross multiplication"

*x*

*y*- 3

*x[i/] = 3*

Is that correct??

*y*- 1 "Will minus both sides with 3*y*"*x**y*- 3*x*- 3*y*= -1 "Will add both sides with 3*x*"*x**y*- 3*y*= -1 + 3*x**y*(*x*- 3) = -1 + 3*x*"Factor out*y*"*y*= (-1 + 3*x*)/(*x*- 3) "Divide both sides with (*x*- 3)"Is that correct??