Question 1125641
find its inverse, and evaluate the following:

{{{f(x)=10-root(3,x-8)}}}

first

{{{f(x)=y}}}

{{{y=10-root(3,x-8)}}}....Interchange the variables:
 
{{{x=10-root(3,y-8)}}}...solve for {{{y}}}

{{{root(3,y-8)=10-x}}}

{{{(root(3,y-8))^3=(10-x)^3}}}

{{{y-8=-x^3 + 30 x^2 - 300 x + 1000}}}

{{{y=-x^3 + 30 x^2 - 300 x + 1000+8}}} , so your inverse is

{{{f^(-1)=-x^3 + 30 x^2 - 300 x + 1008}}} 


{{{f^(-1) (10)=-(10)^3 + 30 (10)^2 - 300 (10) + 1008}}} 

{{{f^(-1) (10)=-(1000) + 30 (100) - 300 (10) + 1008}}} 

{{{f^(-1) (10)=-1000 + 3000  -3000  + 1008}}} 
{{{f^(-1) (10)=-1000  + 1008}}} 
{{{f^(-1) (10)=8}}} 



f^(-1) (11)
{{{f^(-1) (11)=-(11)^3 + 30 (11)^2 - 300 (11) + 1008}}} 

{{{f^(-1) (11)=-(1331) + 30 (121) - 300 (11) + 1008}}} 

{{{f^(-1) (11)=-1331 + 3630  -3300  + 1008}}} 
{{{f^(-1) (11)=-4631  + 4638}}} 
{{{f^(-1) (11)=7}}} 





{{{f^(-1) (12)=-(12)^3 + 30 (12)^2 - 300 (12) + 1008}}} 
{{{f^(-1) (12)=-1728 + 4320 - 3600 + 1008}}} 
{{{f^(-1) (12)= 5328 - 5328 }}} 
{{{f^(-1) (12)= 0 }}}