Q103 / 160 · TS EAMCET 2020 (14 Sep Shift 2) · Mathematics

Define $f(x)=\left\{\begin{array}{ll}1+x, & 0 \leq x \leq 2 \\ 3-x, & 2 < x \leq 3\end{array}\right.$. If $f o f(x)$ is discontinuous at $a$ and $b$ in $[0,3]$ and $a < b$, then $2 a+3 b=$

  • A
    3
  • B
    2
  • C
    6
  • Correct answer
    D
    8

Solution

We have, $\begin{aligned} f(x) & = \begin{cases}1+x, & 0 \leq x \leq 2 \\ 3-x, & 2 < x \leq 3\end{cases} \\ f(f(x)) & = \begin{cases}1+f(x), & 0 \leq f(x) \leq 2 \\ 3-f(x), & 2 < f(x) \leq 3\end{cases} \\ f f(x) & = \begin{cases}1+(1+x), & 0 \leq x \leq 1 \\ 3-(1+x), & 1 < x \leq 2 \\ 1+(3-x), & 2 < x \leq 3\end{cases} \\ f(x) & = \begin{cases}2+x, & 0 \leq x \leq 1 \\ 2-x, & 1 < x \leq 2 \\ 4-x, & 2 < x \leq 3\end{cases} \end{aligned}$ $\therefore f o f(x)$ is discontinuous at $x=1$, and $x=2$ $\begin{aligned} & \therefore \quad a=1, b=2 \\ & 2 a+3 b=2(1)+3(2)=2+6=8 \\ & \end{aligned}$

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

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