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
ANSWERED
asked 2020-10-27
Decide whether the composite functions, \(\displaystyle{f}\circ{g}\) and \(\displaystyle{g}\circ{f}\), are equal to x.
\(\displaystyle{f{{\left({x}\right)}}}={x}^{{3}}+{9}\)
\(\displaystyle{g{{\left({x}\right)}}}={\sqrt[{{3}}]{{{x}-{9}}}}\)

Answers (1)

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.
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