If f(x) = (x - 1) / (x + 1), then find the value of f(2x).
Change the subject of formula from f(x) to that for x.
Answer: If f(x) = (x - 1) / (x + 1),then, the value of f(2x) is equal to (3f(x) + 1) / (f(x) + 3).
Given below is the detailed solution using a simple operation, known as componendo and dividendo.
Explanation:
Given , f(x) = (x - 1) / (x + 1).
Applying componendo and dividendo on numerator and denominator, we get,
[f(x) + 1] / [f(x) - 1] = 2x/-2
[f(x) + 1] / [f(x) - 1] = -x
==> [f(x) + 1] / [1 - f(x)] = x
Now replacing x by 2x in the initial equation, and using the value of x in terms of f(x), we get,
f(2x) = (2x - 1) / (2x + 1)
= {2[f(x) + 1 / 1 - f(x) ] - 1} / { 2[f(x) + 1 / 1 - f(x)] + 1}
= [2f(x) + 2 -1 + f(x)] / [2f(x) + 2 + 1 - f(x)]
= (3 f(x) + 1) / (f(x) + 3)