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# If $f(x)=(4-(x-7)^3\},then\;f^{-1}(x)$=____________.

Toolbox:
• we define $f^{-1}(x)$ as a function .
• such that $(fof^{-1})(x)=x$ ie $g=f^{-1}$ is the inverse function of f if $fog=gof =I$
$f(x)=4-(x-7)^3$

Let $y=4-(x-7)^3$

$(x-7)^3=4-y$

$x=(4-y)^{1/3}+7$

we define a function g(y) such that $g(y)=(4-y)^{1/3}+7$

we see that $(fog)(y)=f((4-y)^{1/3}+7)$

$=4-((4-y)^{1/3}+7-7)$

$=4-((4-y)^{1/3})^3$

$=4-(4-y)$

$=y$

Hence y is inverse of f

and $g(x)=f^{-1}(x)=(4-x)^{1/3}+7$