Find the inverse of the function

f(x) = x / 2x+1

I know that you would make this
x= y / 2x+1 but now I do not know what to do.

To find the inverse of a function, you need to interchange the roles of x and y and then solve for y. Let's start with the function f(x) = x / (2x + 1).

Step 1: Replace f(x) with y:
y = x / (2x + 1)

Step 2: Interchange x and y:
x = y / (2y + 1)

Step 3: Solve for y:
Multiply both sides of the equation by (2y + 1):
x(2y + 1) = y

Expand the left side of the equation:
2xy + x = y

Isolate the variable y on one side and all other terms on the other side:
2xy - y = -x
y(2x - 1) = -x

Finally, divide both sides by (2x - 1) to solve for y:
y = -x / (2x - 1)

Thus, the inverse of the function f(x) = x / (2x + 1) is given by:
f^(-1)(x) = -x / (2x - 1)