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