Question

If $F:R \to R$ is given by $f(x) = {\left( {3 - {x^{\dfrac{1}{3}}}} \right)^{\dfrac{1}{3}}}$ find fofx.

Answer 1

Here we have to place the given function itself in place of x in the given sum to find fofx. So putting f(x) in place of x solves the problem.

Complete step-by-step answer:
Given that,
$f(x) = {\left( {3 - {x^{\dfrac{1}{3}}}} \right)^{\dfrac{1}{3}}}$

$$fof(x) \Rightarrow f\left( {f\left( x \right)} \right) = {\left( {3 - {{\left( {f\left( x \right)} \right)}^3}} \right)^{\dfrac{1}{3}}} \\\ \Rightarrow {\left( {3 - {{\left( {{{\left( {3 - {x^3}} \right)}^{\dfrac{1}{3}}}} \right)}^3}} \right)^{\dfrac{1}{3}}} \\\ \Rightarrow {\left( {3 - {{\left( {3 - {x^3}} \right)}^1}} \right)^{\dfrac{1}{3}}} \\\ \Rightarrow {\left( {3 - 3 + {x^3}} \right)^{\dfrac{1}{3}}} \\\ \Rightarrow {\left( {{x^3}} \right)^{\dfrac{1}{3}}} \\\ \Rightarrow x \\\$$

The value of $fof(x)$= $x$.

Additional information:
Relation defines a relationship between a set of values of pairs.
There are various types of relations.
1.Void Relation
When there is no element of a set related to any element of the same set then it is called a void set.
2. Identity Relation
It is a set in which every element of a set is related to every element of the set.
3.Transitive Relation
If a and b belongs to relation R. b and c belongs to R. and a and c also belongs to R then there is a transitive relation between a, b and c of a set.
4.Reflexive Relation
It is a relation in which every element of a set is related to itself.

Note: Students just need to replace the value of x by f(x) in the given function, and solve further to get an answer.