Find domain of fog, if 1. f(x)=x+5. g(x)=7/(x+7) 2. f(x)=sqrtx. g(x)=6x+18

Chesley 2021-09-09 Answered
Find domain of fog, if
1. \(\displaystyle{f{{\left({x}\right)}}}={x}+{5}.{g{{\left({x}\right)}}}=\frac{{7}}{{{x}+{7}}}\)
2. \(\displaystyle{f{{\left({x}\right)}}}=\sqrt{{x}}.{g{{\left({x}\right)}}}={6}{x}+{18}\)

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Expert Answer

2abehn
Answered 2021-09-10 Author has 13313 answers

1. Replace \(x\) by \(\displaystyle g{{\left({x}\right)}}\in f{{\left({x}\right)}}\):
\(\displaystyle{f{{\left({x}\right)}}}={x}+{5},{g{{\left({x}\right)}}}=\frac{{7}}{{{x}+{7}}}\)
\(\displaystyle{\left({f}{o}{g}\right)}{\left({x}\right)}={f{{\left({g{{\left({x}\right)}}}\right)}}}={f{{\left(\frac{{7}}{{{x}+{7}}}\right)}}}\)
\(\displaystyle=\frac{{7}}{{{x}+{7}}}+{5}\)
\(\displaystyle=\frac{{{7}+{5}{x}+{35}}}{{{x}+{7}}}\)
\(\displaystyle{\left({f}{o}{g}\right)}{\left({x}\right)}=\frac{{{5}{x}+{42}}}{{{x}+{7}}}\)
\(\displaystyle{x}+{7}={0}\)
\(\displaystyle{x}=-{7}\)
Domain \(\displaystyle{\left({f}{o}{g}\right)}={\left(-\infty,-{7}\right)}\cup{\left(-{7},\infty\right)}\)
2. \(\displaystyle{f{{\left({x}\right)}}}=\sqrt{{x}}.{g{{\left({x}\right)}}}={6}{x}+{18}\)
\(\displaystyle{\left({f}{o}{g}\right)}{\left({x}\right)}={f{{\left({g{{\left({x}\right)}}}\right)}}}\)
\(\displaystyle={f{{\left({6}{x}+{18}\right)}}}\)
\(\displaystyle=\sqrt{{{6}{x}+{18}}}\)
\(\displaystyle{\left({f}{o}{g}\right)}{\left({x}\right)}={f{{\left({g{{\left({x}\right)}}}\right)}}}\)
\(\displaystyle=\sqrt{{{6}{x}+{18}}}\)
\(\displaystyle{6}{x}+{18}\ge{0}\)
\(\displaystyle{6}{x}\ge-{18}\)
\(\displaystyle{x}\ge-{3}\)
Domain \(\displaystyle{\left({f}{o}{g}\right)}={\left[-{3},\infty\right)}\)

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