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Homework Help: Inverse of function

  1. Apr 1, 2008 #1
    1. The problem statement, all variables and given/known data

    f: R -> R defined by f(x) = x^3 + 1

    a) determine if it is one to one
    b)find its inverse
    c) calculate (f o f)

    2. Relevant equations

    if f(x) is one to one, if a = b, f(a) = f(b)

    3. The attempt at a solution

    a) yes it is because a^3 + 1 can't equal to b^3 + 1 unless a = b, by definition, this function is onto
    b)the inverse is f^-1 (x) = (x - 1)^(1/3)
    c) f o f = f(f(x) = f(x^3 + 1) = (x^3 + 1)^3 + 1

    just want to make sure i got this, thanks :)
  2. jcsd
  3. Apr 1, 2008 #2
    For #1, by definition the function is 1-1 since you just said that if f(a) = f(b) then a = b, not onto. Onto means that if you have something in your codomain, b in R, then there's an element in your domain, a in R, such that f(a) = b for every b in R in the mapping F: R -> R.

    For #2, let's see. If you have f(x) = x^3 + 1, then using basic manipulations, you get (y-1)^(1/3) = x if you let y = f(x). Then, change this to f^-1(x) = (x-1)^(1/3), which is what you got.

    For #3, you got it. In this case, your x is equal to x^3+1 in F(x). Therefore, if F(x) = x^3 +1, then F(x^3 + 1) = (x^3 + 1)^3 + 1, which is what you also got.

    Re-write #1 and all is good.
    Last edited: Apr 1, 2008
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