Asked by Will
The function
f(x) = cx/2x+3
satisfies f(f(x))=x for all real numbers x\= -3/2. Find c.
f(x) = cx/2x+3
satisfies f(f(x))=x for all real numbers x\= -3/2. Find c.
Answers
Answered by
Steve
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
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