The function f(x) = cx/2 x+3 satisfies f(f(x))=x for all real numbers x\= -3/2. Find c.

Since referring you to the related questions seems to do no good, here it is again:

f(x) = cx/(2x+3)
f(f) = cf/(2f+3) = c[cx/(2x+3)]/(2[cx/(2x+3)]+3)
= c^2x/(2(c+3)x+9)

So, if
c^2x/(2(c+3)x+9) = x,
c^2x = 2(c+3)x^2 + 9x
2(c+3)x^2 + (9-c^2)x = 0
x(2c+6+9-c^2) = 0
x(15+2c-c^2) = 0
x(5c-1)(3c+1) = 0
So, c = 1/5 or c = -1/3

To find the value of c, let's follow these steps to solve the equation f(f(x)) = x:

Step 1: Substitute f(x) into the equation:
f(f(x)) = x
f(cx/2 + 3) = x

Step 2: Replace x with the expression cx/2 + 3 in the equation:
f(cx/2 + 3) = cx/2 + 3

Step 3: Replace f(x) with the expression cx/2 + 3 in the equation:
c((cx/2 + 3)/2) + 3 = cx/2 + 3

Step 4: Simplify the equation:
(c^2x + 6c + 18)/4 + 3 = cx/2 + 3

Step 5: Multiply both sides of the equation by 4 to remove the denominator:
c^2x + 6c + 18 + 12 = 2cx + 12

Step 6: Rearrange the equation to have all the terms on one side:
c^2x - 2cx + 6c + 18 - 12 = 0

Step 7: Combine like terms:
c^2x - 2cx + 6c + 6 = 0

Step 8: Factor out the common term of c:
c(x - 2) + 6(x + 1) = 0

Step 9: Set each term equal to zero:
x - 2 = 0 --> x = 2
x + 1 = 0 --> x = -1

Given that x cannot be -3/2, we can discard -1 as a solution.

Step 10: Substitute x = 2 back into the equation to solve for c:
2 - 2 = 0
0 + 6c + 6 = 0
6c + 6 = 0
6c = -6
c = -1

Therefore, the value of c is -1.