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Question: If \(f\left( x \right) = {\left( {p - {x^n}} \right)^{\dfrac{1}{n}}},p > 0\) and n is a positive int...

If f(x)=(p−xn)1n,p>0f\left( x \right) = {\left( {p - {x^n}} \right)^{\dfrac{1}{n}}},p > 0 and n is a positive integer, then f(f(x))f\left( {f\left( x \right)} \right)=?
A. x
B. xn{x^n}
C. p1n{p^{\dfrac{1}{n}}}
D. p−xnp - {x^n}

Explanation

Solution

Start by writing the given function f(x) and find out f(f(x))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(x)=(p−xn)1n,p>0f\left( x \right) = {\left( {p - {x^n}} \right)^{\dfrac{1}{n}}},p > 0
Let us find out the value of f(f(x))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(f(x))=(p−((p−xn)1n)n)1nf\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 1n\dfrac{1}{n} and n gets cancelled and on simplification, we have
f(f(x))=(p−(p−xn))1n ⇒f(f(x))=(p−p+xn)1n  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(f(x))=(xn)1nf\left( {f\left( x \right)} \right) = {\left( {{x^n}} \right)^{\dfrac{1}{n}}}
Now, Again the powers of n and 1n\dfrac{1}{n} will be cancelled out and hence we have
f(f(x))=xf\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[f(f(x))]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.