Functions 2 Question 17
18. If $f(x)=\left(a-x^{n}\right)^{1 / n}$, where $a>0$ and $n$ is a positive integer, then $f[f(x)]=x$.
(1983, 1M)
Analytical & Descriptive Questions
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Solution:
- Given,
$$ f(x)=\left(a-x^{n}\right)^{1 / n} $$
$$ \begin{array}{ll} \Rightarrow & f[f(x)]=\left[a-{\left(a-x^{n}\right)^{1 / n} }^{n}\right]^{1 / n}=\left(x^{n}\right)^{1 / n}=x \\ \therefore & f[f(x)]=x \end{array} $$
Hence, given statement is true.
