Question

If $ f(x) = 2x $ and $ f(f(x)) = x + 1 $ , then the value of $ x $ is
(a) $ \dfrac{1}{3} $
(b) $ 1 $
(c) $ 2 $
(d) $ 3 $
(e) $ 5 $

Answer 1

Hint: AS we know that the above question consists of a functional equation. A functional equation is any equation in which the unknown represents the function. We know that this type of function assigns exactly one output to each specified type. It is common to name the functions $ f(x) $ or $ g(x) $ . These functional equations have a common technique for solving the value of $ f(x) $ . A function $ f(x) $ is known as a continuous function.

Complete step-by-step answer:
As per the given question we have $ f(x) = 2x $ and $ f(f(x)) = x + 1 $ .
We will take the second equation and then simplify it in the terms of the first equation: $ f(f(x)) = x + 1 $ , we can see that the function $ f $ is twice so we can write it as
  $ 2[f(x)] = x + 1 $ ,
Also we have been given the value of $ f(x) $ is $ 2x $ .
So by substituting the value, we can write it as:
 $ 2(2x) = x + 1 $ .
Now we solve for $ x $ ,
 $ 4x = x + 1 \\
\Rightarrow 4x - x = 1 $
Therefore the required value of $ x $ is $ \dfrac{1}{3} $ .
Hence the correct option is (a) $ \dfrac{1}{3} $ .
So, the correct answer is “Option a”.

Note: Before solving this type of question we should the function equation, their formulas and method to solve it. We should also have the knowledge of the algebraic identities as they are very useful in calculation of this kind of problem. We should note that a function is a relation in which each input has only one output.