Question

Find and simplify in expression for the idicated composite functions. State the domain using interval notation.
\(\displaystyle{f{{\left({x}\right)}}}={3}{x}-{1}\)
\(\displaystyle{g{{\left({x}\right)}}}=\frac{{1}}{{{x}+{3}}}\)
Find \(\displaystyle{\left({g}\circ{f}\right)}{\left({x}\right)}\)

Answer 1

Obtain the composite function, \(\displaystyle{g}\circ{f{{\left({x}\right)}}}\)
\(\displaystyle{g}\circ{f{{\left({X}\right)}}}={g{{\left({f{{\left({x}\right)}}}\right)}}}\)
\(\displaystyle=\frac{{1}}{{{\left({f{{\left({x}\right)}}}\right)}+{3}}}\)
\(\displaystyle=\frac{{1}}{{{\left({3}{x}-{1}\right)}+{3}}}\)
\(\displaystyle=\frac{{1}}{{{3}{x}+{2}}},{x}\ne-\frac{{2}}{{3}}\)
Here, the simplified expression for the composite function is
\(\displaystyle{g}\circ{f{{\left({x}\right)}}}=\frac{{1}}{{{3}{x}+{2}}},{x}\ne-\frac{{2}}{{3}}\)
The composite function defined for all values of x except at \(\displaystyle{x}=-\frac{{2}}{{3}}\) that is defined all values on the left and right of \(\displaystyle{x}=-\frac{{2}}{{3}}\) but not at \(\displaystyle{x}=-\frac{{2}}{{3}}\), thus the domain in the interval notation becomes \(\displaystyle{\left(-\infty,-\frac{{2}}{{3}}\right)}\cup{\left(\frac{{2}}{{3}},\infty\right)}\)