Find the difference quotient of f, that is, find (f(x+h)−f(x))/h , h≠0, for each function. Be sure to simplify. f(x) = 1/(x+3)

Question
Analyzing functions
asked 2021-01-19
Find the difference quotient of f, that is, find \(\displaystyle\frac{{{f{{\left({x}+{h}\right)}}}−{f{{\left({x}\right)}}}}}{{h}},{h}≠{0}\), for each function. Be sure to simplify. \(\displaystyle{f{{\left({x}\right)}}}=\frac{{1}}{{{x}+{3}}}\)

Answers (1)

2021-01-20
\(\displaystyle{f{{\left({x}\right)}}}=\frac{{1}}{{{x}+{3}}}\)
\(\displaystyle{f{{\left({x}+{h}\right)}}}–{f{{\left({x}\right)}}}\frac{{)}}{{h}}=\frac{{\frac{{1}}{{{x}+{h}+{3}}}–\frac{{1}}{{{x}+{3}}}}}{{h}}=\frac{{\frac{{{x}+{3}–{\left({x}+{3}+{h}\right)}}}{{{\left({x}+{h}+{3}\right)}{\left({x}+{3}\right)}}}}}{{h}}={\left(\frac{{{x}+{3}–{x}–{3}–{h}}}{{{\left({x}+{h}+{3}\right)}{\left({x}+{3}\right)}}}\right)}{\left(\frac{{1}}{{h}}\right)}={\left(\frac{{-{h}}}{{{\left({x}+{h}+{3}\right)}{\left({x}+{3}\right)}}}\right)}{\left(\frac{{1}}{{h}}\right)}=\frac{{-{1}}}{{{\left({x}+{h}+{3}\right)}{\left({x}+{3}\right)}}}\)
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