Let f(x)=x+1 and g(x)=2x. Also denote the inverses to these functions as f^-1 and g^-1. Compute \( \[f(g^{-1}(f^{-1}(f^{-1}(g(f(5)))))).\]\)
find f^-1 = y = x+1 solve for x and then switch x and y's x = y-1 so f^-1 = y = x-1
find g^-1 y = 2x solve for x x = y/2 switch y's and x's y = x/2 = g^-1
Now f(5) (this is the original f(x) function....not thte inverse) f(5) = 5+1 = 6
then g(6) = 2x = 2(6) = 12
then f^-1 (12) = 11
then f^-1 (11) = 10
then g^-1 (10 ) = 5
then f (7) = 6
+36916 Problems posting (internal server error) ... corrected last step in answer below.... ~EP
+36916 find f^-1 = y = x+1 solve for x and then switch x and y's x = y-1 so f^-1 = y = x-1
find g^-1 y = 2x solve for x x = y/2 switch y's and x's y = x/2 = g^-1
Now f(5) (this is the original f(x) function....not thte inverse) f(5) = 5+1 = 6
then g(6) = 2x = 2(6) = 12
then f^-1 (12) = 11
then f^-1 (11) = 10
then g^-1 (10 ) = 5
then f (5) = 6
