Gujarat BoardEnglish MediumSTD 11 ScienceMATHSFunctions4 Marks
Question
If $\text{f(x)}=\log_\text{e}(1-\text{x})$ and $\text{g(x)}=[\text{x}],$ then determine the following functions: fg
✓
Answer
We have, $\text{f(x)}=\log_\text{e}(1-\text{x})$ and $\text{g(x)}=[\text{x}]$ $\text{f(x)}=\log_\text{e}(1-\text{x})$ is defined, if 1 - x > 0 $\Rightarrow1>\text{x}$ $\Rightarrow\text{x}<1$ $\Rightarrow\text{x}\in-\infty,1)$ $\therefore\text{ Domain(f)}=(-\infty,1)$ $\text{g(x)}=[\text{x}]$ is defined for all $\text{x}\in\text{R}$ $\therefore\ \text{Domain(g)}=\text{R}$ $\therefore\ \text{Domain(f)}\cap\text{R}\text{ Domain(g)}=(-\infty,1)\cap\text{R}$ $=(-\infty,1)$ $\text{f}+\text{g}:(-\infty,1)\rightarrow\text{R}$ defined by $(\text{fg)(x)}=\text{f(x)}\times\text{g(x)}$ $=\log_\text{e}(1-\text{x})\times[\text{x}]$ $=[\text{x}]\log_\text{e}(1-\text{x})$
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