Find the inverse of f for which f is defined as f:x-(2x minus 1)divided by (2x plus 1)
To find the inverse of a function, we need to switch the roles of x and f(x) and solve for x. Let's denote the inverse function as g(x).
Step 1: Replace f(x) with y.
f(x) = y
Step 2: Swap x and y.
x = (2y - 1) / (2y + 1)
Step 3: Solve for y.
Begin by cross-multiplying:
x(2y + 1) = 2y - 1
Distribute x:
2xy + x = 2y - 1
Move all terms containing y to one side:
2xy - 2y = -1 - x
Factor out y:
y(2x - 2) = -1 - x
Divide both sides by (2x - 2):
y = (-1 - x) / (2x - 2)
Step 4: Replace y with g(x).
g(x) = (-1 - x) / (2x - 2)
Therefore, the inverse of f is g(x) = (-1 - x) / (2x - 2).