Question

If $f\left( x \right) = {\left( {a - {x^n}} \right)^{\dfrac{1}{n}}}$ where $a > 0$ and $n \in N$, then $fof\left( x \right)$ is equal to:
(A) $a$
(B) $x$
(C) ${x^n}$
(D) ${a^n}$

Answer 1

Hint: We are required to find the value of a composite function $fof\left( x \right)$ when we are given that $f\left( x \right) = {\left( {a - {x^n}} \right)^{\dfrac{1}{n}}}$ where $a > 0$ and $n \in N$. This question requires us to have the knowledge of basic and simple algebraic rules and operations such as substitution, addition, multiplication, subtraction and many more like these. A thorough understanding of functions and it’s applications can be of great significance.

Complete step-by-step answer:
Here, $f\left( x \right) = {\left( {a - {x^n}} \right)^{\dfrac{1}{n}}}$ such that $a > 0$ and $n \in N$.
In the given question, we are required to find the value of a function for a certain value of a variable. The value of a function at a certain value of variable is found by substituting the value of variable as specified in the question into the function.
So, we need to replace the variable in the function given to us in the question by the value specified.
So, the function given to us is: $f\left( x \right) = {\left( {a - {x^n}} \right)^{\dfrac{1}{n}}}$.
We are required to find the value of $fof\left( x \right)$ by replacing the value of variable x in the function by the function $f\left( x \right) = {\left( {a - {x^n}} \right)^{\dfrac{1}{n}}}$.
Hence, $fof\left( x \right) = f\left[ {f\left( x \right)} \right]$
$ \Rightarrow fof\left( x \right) = f\left[ {{{\left( {a - {x^n}} \right)}^{\dfrac{1}{n}}}} \right]$
Now, putting ${\left( {a - {x^n}} \right)^{\dfrac{1}{n}}}$ into the function $f\left( x \right) = {\left( {a - {x^n}} \right)^{\dfrac{1}{n}}}$ in place of x, we get,
$ \Rightarrow fof\left( x \right) = {\left( {a - {{\left( {{{\left( {a - {x^n}} \right)}^{\dfrac{1}{n}}}} \right)}^n}} \right)^{\dfrac{1}{n}}}$
Simplifying the expression, we get,
$ \Rightarrow fof\left( x \right) = {\left( {a - \left( {a - {x^n}} \right)} \right)^{\dfrac{1}{n}}}$
Opening the brackets,
$ \Rightarrow fof\left( x \right) = {\left( {a - a + {x^n}} \right)^{\dfrac{1}{n}}}$
$ \Rightarrow fof\left( x \right) = {\left( {{x^n}} \right)^{\dfrac{1}{n}}}$
Simplifying the expression further using the law of exponent ${\left( {{x^a}} \right)^b} = {x^{ab}}$, we get,
$ \Rightarrow fof\left( x \right) = x$
Hence, we get the value of the $fof\left( x \right)$ as $x$ by replacing the variable in the original function by the function itself.
So, the correct answer is “Option B”.

Note: Such questions that require just simple change of variable can be solved easily by keeping in mind the algebraic rules such as substitution and transposition. Substitution of a variable involves putting a certain value in place of the variable. That specified value may be a certain number or even any other variable.