+130511 OK.....here we go.... f-1 = x- 2 and g-1 = 3x
So f(19) = (19) + 2 = 21
And we want to put this into g...so we have (21)/3 = 7
Now...we want to put this result into f-1 = (7)-2 = 5
And we want to put this result into f-1 again = (5)-2 = 3
And we want t put this result into g-1 = 3(3) = 9
Finally..we want to put this result into f = (9) + 2 = 11
And there you are.....
+130511 OK.....here we go.... f-1 = x- 2 and g-1 = 3x
So f(19) = (19) + 2 = 21
And we want to put this into g...so we have (21)/3 = 7
Now...we want to put this result into f-1 = (7)-2 = 5
And we want to put this result into f-1 again = (5)-2 = 3
And we want t put this result into g-1 = 3(3) = 9
Finally..we want to put this result into f = (9) + 2 = 11
And there you are.....
![Let $f(x)=x+2$ and $g(x)=x/3.$ Also denote the inverses to these functions as $f^{-1}$ and $g^{-1}.$ Compute \[f(g^{-1}(f^{-1}(f^{-1}(g(f(19)))))).\]](/api/ssl-img-proxy?src=https%3A%2F%2Fdata.artofproblemsolving.com%2Faops20%2Flatex%2Fimages%2Fa3a89c654e04738a92db5e18c336e48ba90b0806.png)
