Given f ( x ) = 3x - 2 find f ^{-1} ( x ).
Solution
Now, already we know what the inverse to this function is as already we've done some work with it. Though, it would be nice to really start with this as we know what we must get. it will work as a nice verification of the procedure.
We'll first replace f ( x ) with y.
y = 3x - 2
Next, replace all x's with y & all y's with x.
x =3 y - 2
Now, solve out for y.
x + 2 = 3 y
1 /3 (x + 2) = y
x/3 + 2/3=y
At last replace y with f ^{-1} ( x ) .
f ^{-1} ( x ) = x/3 + 2/3
Now, we have to verify the results. Already we took care of this in the earlier section; though, we actually should follow the procedure so we'll do that here. It doesn't issue which of the two that we verify we just have to check one of them. This time we'll verify that
( f o f ^{-1} )( x ) = x is true.
( f o f ^{-1} )( x ) = f[ f ^{-1} ( x )]
= f [x/3 + 2 /3]
= 3( x /3 + 2/3 ) - 2
= x + 2 - 2
=