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Question
If f(x) = (a − xn)1/n, a > 0 and n ∈ N, then prove that f(f(x)) = x for all x.
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Solution
Given:
f(x) = (a − xn)1/n, a > 0
Now,
f{ f (x)} = f (a − xn)1/n
= [a – {(a – xn)1/n}n]1/n
= [ a – (a – xn)]1/n
= [ a – a + xn)]1/n = (xn)1/n = x(n × 1/n) = x
Thus, f(f(x)) = x.
Hence proved.
f(x) = (a − xn)1/n, a > 0
Now,
f{ f (x)} = f (a − xn)1/n
= [a – {(a – xn)1/n}n]1/n
= [ a – (a – xn)]1/n
= [ a – a + xn)]1/n = (xn)1/n = x(n × 1/n) = x
Thus, f(f(x)) = x.
Hence proved.
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