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

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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) = 2x + 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.


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Here's the other one:

The function f : R --> R defined by f(x) = (3x - 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) = (3x - 1)/x - 3)
x = (3y - 1)/y - 3) "replace x with y"
x(y - 3) = 1(3y - 1)) "Cross multiplication"
xy - 3x[i/] = 3y - 1 "Will minus both sides with 3y"
xy - 3x - 3y = -1 "Will add both sides with 3x"
xy - 3y = -1 + 3x
y(x - 3) = -1 + 3x "Factor out y"
y = (-1 + 3x)/(x - 3) "Divide both sides with (x - 3)"

Is that correct??
 

Answers and Replies

  • #2
Hurkyl
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This site is not an answer service! We will be happy to help you work through the problem, but this requires you to show what you've done on the problem so far (even if it's just pondering what you can do, or what needs to be done).


I think you stated something incorrectly as well -- the expression you wrote as a definition for f doesn't define a function from R to R.
 

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