Question

# Find and simplify in expression for the idicated composite functions. State the domain using interval notation. f(x)=3x-1 g(x)=1/(x+3) Find ([email protected])(x)

Composite functions
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 $$\displaystyle{\left({g}\circ{f}\right)}{\left({x}\right)}$$

2021-02-26
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 $$\displaystyle{\left(-\infty,-\frac{{2}}{{3}}\right)}\cup{\left(\frac{{2}}{{3}},\infty\right)}$$