Question 1197463
Find inverse of the following function:


{{{f(x)= (x^2+ 9)/(x - 5)}}}   {{{x<>5}}}


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


{{{y= (x^2+ 9)/(x - 5)}}} ........swap variables


{{{x= (y^2+ 9)/(y - 5)}}}...........solve for {{{y}}}


{{{x(y - 5)= y^2+ 9}}}


{{{xy - 5x= y^2+ 9}}}


{{{-5x-9= y^2-xy}}}...complete square on right side


{{{-5x-9= (y^2-xy+(x/2)^2)-(x/2)^2}}}


{{{-5x-9= (y-x/2)^2-x^2/4}}}


{{{x^2/4-5x-9= (y-x/2)^2}}}


{{{sqrt((x^2 - 20 x - 36)/4 )= y-x/2}}}


{{{y=(1/2)sqrt(x^2 - 20 x - 36)+x/2}}}



=> inverse is


{{{f}}}'{{{(x)=(1/2)(x+-sqrt(x^2 - 20x - 36))}}}




{{{ graph( 600, 600, -50, 50, -50, 50, (x^2+ 9)/(x - 5), (1/2)(x+sqrt(x^2 -20x - 36)), (1/2)(x-sqrt(x^2 - 20x - 36))) }}}