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October 4th, 2012, 05:24 PM   #1
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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(x-2);
(b) F(x)=f^2(x);


Can someone please walk me through the steps of answering this question?
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October 5th, 2012, 04:26 AM   #2
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(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) = x, 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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