SOLUTION: consider the function f(x) = (cx)/ (2x + 3) , where x is not ewual to -3/2. Find all values of c (if any) for which f(f(x)) = x

Algebra ->  Test -> SOLUTION: consider the function f(x) = (cx)/ (2x + 3) , where x is not ewual to -3/2. Find all values of c (if any) for which f(f(x)) = x      Log On


   

Question 41949: consider the function f(x) = (cx)/ (2x + 3) , where x is not ewual to -3/2. Find all values of c (if any) for which f(f(x)) = x
Answer by psbhowmick(878) About Me  (Show Source):
You can put this solution on YOUR website!
f%28x%29+=+c%2Ax%2F+%282x+%2B+3%29+
Therefore, f%28f%28x%29%29+=+c%2Af%28x%29%2F+%282%2Af%28x%29+%2B+3%29+

[Just replace 'x' with f(x)]

Thus,
or
or f%28f%28x%29%29+=+c%5E2%2Ax%2F%282%2Ac%2Ax%2B6x%2B9%29

So, if f(f(x)) = x then
c%5E2%2Ax%2F%282%2Ac%2Ax%2B6x%2B9%29+=+x
or c%5E2%2F%282%2Ac%2Ax%2B6x%2B9%29+=+1
or c%5E2+=+2%2Ac%2Ax%2B6x%2B9
or c%5E2-9=2cx%2B6x
or x=%28c%5E2-9%29%2F%282c%2B6%29

As x cannot be equal to -3%2F2
so %28c%5E2-9%29%2F%282c%2B6%29 can also not be equal to -3%2F2.

But, however, if %28c%5E2-9%29%2F%282c%2B6%29+=+-3%2F2 then
2%28c%5E2-9%29%2B3%282c%2B6%29+=+0
or 2c%5E2%2B6c+=+0
or c%28c%2B3%29=0
Hence either c = 0 or c = -3.
Thus f(f(x)) = x is valid for all values of 'x' except c = 0 and c = -3.