Question

if a function \[f\] satisfies \[f\{ f(x)\} = x + 1\] for all real values of x and \[f(0) = \dfrac{1}{2}\], then \[f(1)\]is equal to
1) \[\dfrac{1}{2}\]
2) \[1\]
3) \[\dfrac{3}{2}\]
4) \[2\]
5) \[0\]

Answer 1

Hint: This is a question based on functional equations. For finding the required value of the function, we need to manipulate the given functional equation to our requirements. Here, no perquisite formula is required in this question.

Complete step-by-step answer:
 Let us begin the question with the given functional equation as shown below
\[ \Rightarrow f\{ f(x)\} = x + 1\]
Also, we are given the value of the function f at x=0, as shown below,
\[ \Rightarrow f(0) = \dfrac{1}{2}\]
Therefore, let us proceed with the question by putting x=0 in the functional equation as shown below,
\[ \Rightarrow x = 0\]
Now, after putting x=0, we get,
\[ \Rightarrow f\{ f(0)\} = 0 + 1\]
Now, applying the given value of f(x) at x=0 in the left-hand side of the equation as shown below,
\[ \Rightarrow f(\dfrac{1}{2}) = 1\]
Now, let us observe and understand what we have calculated; after putting x=0 in the function equation, we received the value of the function at x=1/2,
\[ \Rightarrow x = \dfrac{1}{2}\]
Let us proceed in the question by putting x=1/2 in the functional equation as shown below,
\[ \Rightarrow f(f(\dfrac{1}{2})) = \dfrac{1}{2} + 1\]
Now, after replacing the value of f(1/2)=1 in the left-hand side of the equation we get,
\[ \Rightarrow f(1) = \dfrac{3}{2}\]
And in the question, we were required to find the value of the function at x=1,
Therefore, the correct answer is an option(3).
So, the correct answer is “Option 3”.

Note: This is a general question on functional equations. One should be well versed in the concepts of observing and manipulating functional equations. This question has no particular formula but an approach to solving, so keep that in mind. Do not commit calculation mistakes, and be sure of the final answer.