Answer:
The value of [tex]f^{-1}(f(3))[/tex] is 3.
Step-by-step explanation:
Here, the given function is,
[tex]f(x)=9x+1-----(1)[/tex]
For finding [tex]f^{-1}(f(3))[/tex] we have to find out the inverse of f(x),
The steps are as follows,
Step 1 : Replace f(x) by y,
[tex]y=9x+1[/tex]
Step 2 : Switch x and y,
[tex]x=9y+1[/tex]
Step 3 : Isolate y in the left side,
[tex]-9y=1-x[/tex]
[tex]\implies y = \frac{x-1}{9}[/tex]
Step 4 : replace y by [tex]f^{-1}(x)[/tex],
[tex]f^{-1}(x)=\frac{x-1}{9}------(2)[/tex]
Now,
[tex]f^{-1}(f(3))=f^{-1}(9\times 3+1)[/tex] ( From equation (1) )
[tex]=f^{-1}(28)[/tex]
[tex]=\frac{28-1}{9}[/tex] ( From equation (2) )
[tex]=\frac{27}{9}[/tex]
[tex]=3[/tex]
