Q79 / 160 · TS EAMCET 2014 · Mathematics

Let $Q$ be the set of all rational numbers in $[0,1]$ and $f:[0,1] \rightarrow[0,1]$ be defined by $$ f(x)=\left\{\begin{array}{r} x \text { for } x \in Q \\ 1-x \text { for } x \notin Q \end{array}\right. $$ Then, the set $S=\{x \in[0,21]:(f o f)(x)\}$ is equal to

  • Correct answer
    A
    $[0,1]$
  • B
    $-Q$
  • C
    $[0,1]-Q$
  • D
    $(0,1)$

Solution

Given, $f(x)=\left\{\begin{array}{cc}x & \text { for } x \in Q \\ 1-x & \text { for } x \notin Q\end{array}\right.$ is defined for $$ f:[0,1] \rightarrow[0,1] $$ If $x$ is rational, then $$ \begin{array}{rlrl} f(x) & =x \\ \therefore \quad & f(f(x)) & =f(x)=x \end{array} $$ If $x$ is irrational, then $$ \begin{aligned} f(x) & =1-x \\ \therefore \quad f \circ f(x) & =f(1-x)=1-(1-x) \\ & =x \end{aligned} $$ $\therefore f \circ f(x)=x$ is possible for all values of domain $[0,1]$.

Hence, the correct answer is option: 'A'.

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