Given: f(x) = 2x^2+3 and f(v) = 5v+6
We know that f(x) is going to have two inverses; this may cause a problem.
Compute f^-1(x) by substituting f^-1(x) for every x with within f(x):
f(f^-1(x)) = 2(f^-1(x))^2+3
The property of a function and its inverse, f(f^-1(x)) = x, causes the left side to become x:
x = 2(f^-1(x))^2+3
Solve for f^-1(x):
x-3 = 2(f^-1(x))^2
(x-3)/2 = (f^-1(x))^2
f^-1(x) = sqrt((x-3)/2) and f^-1(x) = -sqrt((x-3)/2)
Substitute x = f(v) into the left sides and x = 5v+6 into the right sides:
f^-1(f(v)) = sqrt((5v+6-3)/2) and f^-1(f(v)) = -sqrt((5v+6-3)/2)
Use the property f^-1(f(v)) = v to make the left sides become v:
v = sqrt((5v+6-3)/2) and v = -sqrt((5v+6-3)/2)
We are about to square both equations but this will eliminate, therefore, the two equations degenerate into a single equation:
v^2 = (5v+6-3)/2
2v^2= 5v+3
2v^2- 5v-3=0
Factor:
(2v+1)(v-3) = 0
v = -1/2 and v=3
