Find a formula for the inverse of the function f(x) = (4 x-1)/(2x+3)
step 1. interchange the x and y variables
to get x = (4y-1)/(2x+3)
step 2. Solve this new equation for y
2xy + 3x = 4y-1
2xy - 4y = -3x - 1
y(2x-4) = -3x-1
y = (-3x-1)/(2x-4) or
y = (3x+1)/(4-2x) or
f^-1(x) = (3x+1)/(4-2x)
I usually test my answer by picking any value for x, let's say x=1
f(1) = 3/5 = .6
then f^-1(.6) = 2.8/2.8 = 1
Then inverse should always yield the original input value, that is why we call it the "inverse".
Checking for one value does of course not prove that my answer is right, but the probability of being wrong would be very very small