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If $f\left( x \right) = {\left( {p - {x^n}} \right)^{\dfrac{1}{n}}},p > 0$ and n is a positive integer, then $f\left( {f\left( x \right)} \right)$=?
A. x
B. ${x^n}$
C. ${p^{\dfrac{1}{n}}}$
D. $p - {x^n}$

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Last updated date: 17th Jun 2024
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Answer
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Hint: Start by writing the given function f(x) and find out $f\left( {f\left( x \right)} \right)$by substituting the value of f(x) in place of x , Simplify the new expression formed by solving the exponents and get the most simplified form , the value obtained is the desired answer.

Complete step-by-step answer:
Given,
$f\left( x \right) = {\left( {p - {x^n}} \right)^{\dfrac{1}{n}}},p > 0$
Let us find out the value of $f\left( {f\left( x \right)} \right)$
Substituting the value of f(x), by replacing the equation f(x) in place of x variable , we get
\[f\left( {f\left( x \right)} \right) = {\left( {p - {{\left( {{{\left( {p - {x^n}} \right)}^{\dfrac{1}{n}}}} \right)}^n}} \right)^{\dfrac{1}{n}}}\]
Here , The inside powers $\dfrac{1}{n}$ and n gets cancelled and on simplification, we have
$
  f\left( {f\left( x \right)} \right) = {\left( {p - \left( {p - {x^n}} \right)} \right)^{\dfrac{1}{n}}} \\
   \Rightarrow f\left( {f\left( x \right)} \right) = {\left( {p - p + {x^n}} \right)^{\dfrac{1}{n}}} \\
$
So now p will be cancelled out with -p and we are left with
$f\left( {f\left( x \right)} \right) = {\left( {{x^n}} \right)^{\dfrac{1}{n}}}$
Now, Again the powers of n and $\dfrac{1}{n}$ will be cancelled out and hence we have
$f\left( {f\left( x \right)} \right) = x$

So, the correct answer is “Option A”.

Note: Similar questions can be asked with multiple iteration of f(x) ,for .e.g. $f\left[ {f\left( {f\left( x \right)} \right)} \right]$, follow the same procedure as above. Attention must be given while substituting and simplifying as any missed sign or wrong interpretation may lead to wrong answers.