Check if the following functions have an inverse function. If yes…

Question

Check if the following functions have an inverse function. If yes, find the inverse function.

$f(x)=9 x^3+8$

✓

Answer

$f(x) 9 x^3+8$
Let $\mathrm{f}\left(\mathrm{x}_1\right)=\mathrm{f}\left(\mathrm{x}_2\right)$
$
\therefore 9 x_1^3+8=9 x_2^3+8
$
$
\therefore \mathrm{x}_1=\mathrm{x}_2
$
$\therefore \mathrm{f}$ is a one-one function.
$
\therefore \mathrm{f}(\mathrm{x})=9 \mathrm{x}^3+8=\mathrm{y} \text {, (say) }
$
$\therefore \mathrm{x}=\sqrt[3]{\frac{y-8}{9}}$
$\therefore$ For every y we can get $x$.
$\therefore \mathrm{f}$ is an onto function.
$
\therefore \mathrm{x}=\sqrt[3]{\frac{y-8}{9}}=\mathrm{f}^{-1}(\mathrm{y})
$
Replacing y by $x$, we get
$
\mathrm{f}^{-1}(\mathrm{x})=\sqrt[3]{\frac{x-8}{9}}
$

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