1. Doubt on Math and Stat.


Dear Sir,
Could you please support for clearing my doubts:
1. Math: How to find a domain & range of fog,(x) , gof(x) can you help me.
2. stat: How to identify when to use addition and multiplication rule, Npr, ncr. Means tricky questions the process of simplifications.
Regards
Ayyakutti A

\[\begin{gathered}
{\text{Let us understand this with an example:}} \hfill \\
f(x) = \frac{1}{{1 + x}} \hfill \\
g(x) = \ln x \hfill \\
fog(x) = f[g(x)] = f[\ln x] = \frac{1}{{1 + (\ln x)}} \hfill \\
({\text{Put }}g(x){\text{ in place of }}x{\text{ in }}f(x){\text{, you will get }}fog(x)) \hfill \\
{\text{To find domain, you need to calculate the set of all values for which the function }} \hfill \\
{\text{is defined}}{\text{.}} \hfill \\
{\text{But in exam, just try to find out the value of x at which the function is not defined}}{\text{.}} \hfill \\
fog(x) = \frac{1}{{1 + (\ln x)}} \hfill \\
{\text{Here, }}fog(x){\text{ is not defined when denominator }} = 0{\text{, and when }}\ln x{\text{ is not defined}} \hfill \\
\ln x{\text{ is not defined for negative value of x, hence }}x{\text{ }} > 0 \hfill \\
{\text{denominator of }}fog(x){\text{ can not be zero, hence x}} \ne \frac{1}{e} \hfill \\
{\text{hence domain = all positive value except }}\frac{1}{e} \hfill \\
{\text{Similarly you can calculate the range, it is set of all possible value of outcome for }} \hfill \\
{\text{the domain as input to function}}{\text{.}} \hfill \\
\hfill \\
{\text{Tip for Exam: Use online calculator to eliminate wrong options}}{\text{.}} \hfill \\
\end{gathered} \]
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