Find the inverse of 2/2x-1 and verify.
y = 2/(2x-1)
2x-1 = 2/y
2x = 1 + 2/y
x = 1/2 + 1/y
so, the inverse is 1/2 + 1/x = (x+2)/2x
and the verification?
I kinda figure you would take care of that yourself...
you want f(f^-1(x)) = x
so, plug in (x+2)/2x into the original equation and make sure you wind up with just x.
2/(2[(x+2)/2x]-1)
= 2/((x+2)/x - 1)
= 2/((x+2-x)/x)
= 2/(2/x)
= 2*x/2
= x
To find the inverse of a function, we need to swap the roles of the input and output variables. In this case, we have the function y = 2/(2x - 1).
Step 1: Replace 'y' with 'x' and vice versa to obtain the equation in terms of the inverse variable, let's call it 'w':
x = 2/(2w - 1)
Step 2: Solve the equation for 'w'. To do this, we can start by multiplying both sides of the equation by (2w - 1):
x(2w - 1) = 2
Step 3: Expand and simplify the equation:
2wx - x = 2
Step 4: Move the x term to the other side of the equation:
2wx = x + 2
Step 5: Divide both sides of the equation by 2x:
w = (x + 2) / (2x)
Now we have the equation for the inverse function, which is w = (x + 2) / (2x).
To verify that this is the correct inverse, we can substitute 'w' back into the original equation and see if we get back to 'x'.
Original equation: y = 2/(2x - 1)
Step 1: Replace 'y' with 'w':
w = 2/(2x - 1)
Step 2: Solve for 'x':
Multiply both sides by (2x - 1):
w(2x - 1) = 2
Expand and simplify:
2wx - w = 2
Move the w term to the other side:
2wx = w + 2
Divide both sides by 2w:
x = (w + 2) / (2w)
By comparing this equation with the inverse we found earlier, w = (x + 2) / (2x), we can see that they are equivalent.
Therefore, the inverse of the function 2/(2x - 1) is given by w = (x + 2) / (2x).