The function

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

To find the value of c, we need to solve the equation f(f(x)) = x using the given function f(x) = cx/(2x + 3).

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

Step 2: Multiply both sides of the equation by (2x + 3) to eliminate the fraction.
(2x + 3) * f(x) = x * (2x + 3)

Step 3: Expand both sides of the equation.
(2x + 3) * (cx/(2x + 3)) = 2x^2 + 3x

Step 4: Cancel out the (2x + 3) terms on the left side of the equation.
cx = 2x^2 + 3x

Step 5: Rearrange the equation to bring all terms to one side.
2x^2 + (3 - c)x = 0

Step 6: Since this equation should hold true for all real numbers x ≠ -3/2, the coefficients of corresponding powers of x should be equal.
Thus, we have two conditions to satisfy:

Condition 1: The coefficient of x^2 should be zero.
2 = 0

This condition cannot be satisfied; therefore, there is no value of c that satisfies the given equation f(f(x)) = x for all real numbers x ≠ -3/2.

To find the value of c, we need to substitute f(x) into the equation f(f(x))=x and then solve for c.

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

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

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

Step 4: Simplify further by finding a common denominator:
f(cx/2x+3) = (cx^2 + 6cx + 12x)/(4x^2 + 12x)

Step 5: Set this expression equal to x:
(cx^2 + 6cx + 12x)/(4x^2 + 12x) = x

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

Step 7: Distribute and simplify:
cx^2 + 6cx + 12x = 4x^3 + 12x^2

Step 8: Rearrange the equation to make it equal to zero:
4x^3 + 12x^2 - cx^2 - 6cx - 12x = 0

Step 9: Factor out an x from each term in the equation:
x(4x^2 + 12x - cx - 6 - 12) = 0

Step 10: Combine like terms:
x(4x^2 + (12 - c)x - 18) = 0

Step 11: Since this equation must be equal to zero for all x ≠ -3/2, the expression inside the parentheses must be equal to zero:
4x^2 + (12 - c)x - 18 = 0

Step 12: Compare the equation with the general quadratic equation ax^2 + bx + c = 0. We can see that a = 4, b = (12 - c), and c = -18.

Step 13: For a quadratic equation to have a unique solution, the discriminant (b^2 - 4ac) must be equal to zero.
Therefore, we can write the discriminant equation:
(12 - c)^2 - 4(4)(-18) = 0

Step 14: Simplify and solve for c:
144 - 24c + c^2 + 288 = 0
c^2 - 24c + 432 = 0

Step 15: Factor the quadratic equation:
(c - 18)(c - 24) = 0

Step 16: Solve for c by setting each factor equal to zero:
(c - 18) = 0 or (c - 24) = 0
c = 18 or c = 24

Therefore, the value of c can be either 18 or 24.

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