Question

Decide whether the composite functions, f@g and g@f, are equal to x. f(x)=x^3+9 g(x)=root(3)(x-9)

Composite functions
Decide whether the composite functions, $$\displaystyle{f}\circ{g}$$ and $$\displaystyle{g}\circ{f}$$, are equal to x.
$$\displaystyle{f{{\left({x}\right)}}}={x}^{{3}}+{9}$$
$$\displaystyle{g{{\left({x}\right)}}}={\sqrt[{{3}}]{{{x}-{9}}}}$$

2020-10-28
To find the composite function we need to plug-in function in place of the variable. Mathematically,
$$\displaystyle{f}\circ{g}={f{{\left({g{{\left({x}\right)}}}\right)}}}$$
$$\displaystyle{g}\circ{f}={g{{\left({f{{\left({x}\right)}}}\right)}}}$$
Therefore, the composite functions are,
$$\displaystyle{f}\circ{g}={f{{\left({\sqrt[{{3}}]{{{x}-{9}}}}\right)}}}$$
$$\displaystyle={\sqrt[{{3}}]{{{x}-{9}}}}+{9}$$
$$\displaystyle={x}-{9}+{9}$$
$$\displaystyle={x}$$
$$\displaystyle{g}\circ{f}={g{{\left({f{{\left({x}\right)}}}\right)}}}$$
$$\displaystyle={\sqrt[{{3}}]{{{\left({x}^{{3}}+{9}\right)}-{9}}}}$$
$$\displaystyle={\sqrt[{{3}}]{{{x}^{{3}}}}}$$
=x
Hence, both composite functions are equal to zero.