Mathematics - Functions Question with Solution | TestHub

$\mathrm{y}=\mathrm{f}(\mathrm{x})={\left(5-(\mathrm{x}-8{)}^5\right)}^{1/3}$, then $\begin{array}{rlr}{\mathrm{y}}^3& =5-(\mathrm{x}-8{)}^5\Rightarrow (\mathrm{x}-8{)}^5=5-{\mathrm{y}}^3\Rightarrow \mathrm{x}& =8+{\left(5-{\mathrm{y}}^3\right)}^{1/5}\end{array}$ Let, $\mathrm{z}=\mathrm{g}(\mathrm{x})=8+{\left(5-{\mathrm{x}}^3\right)}^{1/5}$ $\begin{array}{rlr}& \text{ Now, }f(g(x))={\left[5-(x-8{)}^5\right]}^{1/3}& ={\left(5-{\left[{\left(5-x^3\right)}^{1/5}\right]}^5\right)}^{1/3}={\left(5-5+x^3\right)}^{1/3}=x\end{array}$ Similarly, we can show that $\mathrm{g}(\mathrm{f}(\mathrm{x}))=\mathrm{x}$.

Hence, $\mathrm{g}(\mathrm{x})=8+{\left(5-{\mathrm{x}}^3\right)}^{1/5}$ is the inverse of $\mathrm{f}(\mathrm{x})$.