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October 4th, 2012, 05:24 PM  #1 
Newbie Joined: Sep 2012 Posts: 15 Thanks: 0  Let f be an invertible function, and g = f^1....
Let 'f' be an invertible function, and g=f^1. Find which of the following functions F is also invertible (explain, derive a formula for F^1), or give a counter example (can be a graph) if not always so. (a) F(x)=2f(x2); (b) F(x)=f^2(x); Can someone please walk me through the steps of answering this question? 
October 5th, 2012, 04:26 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 18,142 Thanks: 1417 
(a) Inverse is g(x/2) + 2 (since F(g(x/2) + 2) = 2f(g(x/2)) = 2(x/2) = x). (b) Counterexample: f(x) = x³, g(x) = ?x, where x is real, since x^6 is not invertible (1^6 = (1)^6). To obtain the answer for (a), I used the following steps, where G(x) is the inverse to be found (assuming it exists): x = 2f(G(x)  2) f(G(x)  2) = x/2 G(x)  2 = g(x/2) G(x) = g(x/2) + 2. 

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