The function
f(x) = cx/2x+3
satisfies f(f(x))=x for all real numbers x\= -3/2. Find c.
(\= does not equal)
see related question below
To find the value of c, we need to substitute f(x) into f(f(x)) and solve for c.
Given that f(x) = cx/2x + 3, we can substitute f(x) into itself to get f(f(x)) as follows:
f(f(x)) = c(f(x))/(2f(x)) + 3
Substituting the expression for f(x) into the above equation, we have:
f(f(x)) = c(cx/2x + 3)/(2(cx/2x + 3)) + 3
Simplifying the equation further:
f(f(x)) = c^2x/(4x^2) + 3c/(4x) + 3
To satisfy f(f(x)) = x for all real numbers, we need:
c^2x/(4x^2) + 3c/(4x) + 3 = x
Multiplying both sides of the equation by 4x^2 gives us:
c^2x^2 + 3cx + 12x^2 - 4x^3 = 4x^3
Rearranging the equation:
4x^3 - (c^2 - 12)x^2 - 3cx = 0
Since this equation needs to hold true for all x ≠ -3/2, we can equate the coefficients of each term to zero:
4 = 0 (coefficient of x^3)
c^2 - 12 = 0 (coefficient of x^2)
-3c = 0 (coefficient of x)
Solving the equations:
c^2 - 12 = 0
c^2 = 12
c = ±√12
c = ±2√3
Therefore, the values of c that satisfy f(f(x)) = x for all x ≠ -3/2 are c = 2√3 and c = -2√3.