Prove that

Question:
Let $f: R \rightarrow R: f(x)=\left(3-x^{3}\right)^{1 / 3}$. Find $f$ o $f$.

Solution:

To find: $f$ of

Formula used: (i) $f$ o $f=f(f(x))$

Given: (i) $f: R \rightarrow R: f(x)=\left(3-x^{3}\right)^{1 / 3}$

We have,

$f \circ f=f(f(x))=f\left(\left(3-x^{3}\right)^{1 / 3}\right)$

$f \circ f=\left[3-\left\{\left(3-x^{3}\right)^{1 / 3}\right\}^{3}\right]^{1 / 3}$

$=\left[3-\left(3-x^{3}\right)\right]^{1 / 3}$

$=\left[3-3+x^{3}\right]^{1 / 3}$

$=\left[x^{3}\right]^{1 / 3}$

$=x$

Ans) $f$ o $f(x)=x$