Let $f(x) = \sin x$ and $g(x) = \ln |x|$. If the ranges of the composite functions $fog$ and $gof$ are $R_1$ and $R_2$ respectively,then:
- A$R_1 = \{ u: - 1 < u < 1\} , R_2 = \{ v: - \infty < v < 0\}$
- B$R_1 = \{ u: - \infty < u \le 0\} , R_2 = \{ v: - 1 \le v \le 1\}$
- C$R_1 = \{ u: - 1 < u < 1\} , R_2 = \{ v: - \infty < v < 0\}$
- D$R_1 = \{ u: - 1 \le u \le 1\} , R_2 = \{ v: - \infty < v \le 0\}$
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Let $f : R \rightarrow R$ and $g : R \rightarrow R$ be defined as $f(x) = \begin{cases} x+a, & x < 0 \\ |x-1|, & x \geq 0 \end{cases}$ and $g(x) = \begin{cases} x+1, & x < 0 \\ (x-1)^2+b, & x \geq 0 \end{cases}$ where $a, b$ are non-negative real numbers. If $(g \circ f)(x)$ is continuous for all $x \in R$,then $a+b$ is equal to ......
DifficultJEE MAIN 2021
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