Question

If \[f(x) = {(a - {x^n})^{(1/n)}}\] where \[a{\text{ }} > {\text{ }}0\] and , then \[fof\left( x \right)\] is equal to
 (a) \[a\]
 (b) \[x\]
 (c) \[{x^n}\]
 (d) \[{a^n}\]

Answer 1

Hint: To get the value of composite function i.e. f(f(x)), we should find the value of \[f\left[ {{{\left( {a - {x^n}} \right)}^{\dfrac{1}{n}}}} \right]\] and then further x tends to function i.e. \[x \to {(a - {x^n})^{(1/n)}}\]. Simplify the equations by using the exponential properties such as \[{({a^m})^n} = {a^{mn}}\] to get the required result.

Formula Used:
\[x \to {(a - {x^n})^{(1/n)}}\]
\[{({a^m})^n} = {a^{mn}}\]

Complete step by step Solution:
 Before proceeding with the problem, we will know about what are composite functions. Given two functions, we can create another by introducing one function inside the other. The steps required to perform this operation are similar to solving a function with a specific value. Such functions are called composite functions.
Given: \[f(x) = {(a - {x^n})^{(1/n)}}\], where \[a{\text{ }} > {\text{ }}0\],
The given equation is written as a function of f(x)
A composite function is usually a function that is placed inside the other function. Functions are constructed by substituting one function with another. It consists of two or more stages. In this question, fof (x) is the evaluation of the function at the value of the same function.
\[f[f(x)] = f\left[ {{{\left( {a - {x^n}} \right)}^{\dfrac{1}{n}}}} \right]\]
To find \[fof\left( x \right)\] , \[x \to {(a - {x^n})^{(1/n)}}\]in the above-given equation
\[f[f(x)] = {\left[ {a - {{\left( {{{\left( {a - {x^n}} \right)}^{\dfrac{1}{n}}}} \right)}^n}} \right]^{\dfrac{1}{n}}}\] --- eq (1)
We know the exponential property, \[{({a^m})^n} = {a^{mn}}\]
Applying the above exponential property to the eq (1)
\[f[f(x)] = {\left[ {a - {{\left( {a - {x^n}} \right)}^{\dfrac{1}{n} \times n}}} \right]^{\dfrac{1}{n}}}\]
\[f[f(x)] = {\left[ {a - {{\left( {a - {x^n}} \right)}^1}} \right]^{\dfrac{1}{n}}}\]
Simplify the above equation
\[{\text{f}}[{\text{f}}({\text{x}})] = {\left[ {{\text{a}} - \left( {{\text{a}} - {{\text{x}}^{\text{n}}}} \right)} \right]^{\dfrac{1}{{\text{n}}}}}\]
\[f[f(x)] = {\left[ {a - a + {x^n}} \right]^{\dfrac{1}{n}}}\]
Further simplifying the above equation to get
\[{\text{f}}[{\text{f}}({\text{x}})] = {\left[ {{{\text{x}}^{\text{n}}}} \right]^{\dfrac{1}{{\text{n}}}}}\]
Applying the exponential property \[{({a^m})^n} = {a^{mn}}\]to the above equation, we get
\[f[f(x)] = {x^{n \times \dfrac{1}{n}}}\]
\[f[f(x)] = {x^1}\]
Thus, \[fof\left( x \right) = x\]

Hence, the correct option is b.

Note: If [f(x)] is the composite function of f (x) and f (x). In this question, f(x) is a function of f(x). The knowledge of exponential properties is essential for solving the question. Care must be taken while opening the brackets to avoid any mistakes.