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Question # Find the domain of the coposite function [email protected](x)=5/(x+9)g(x)=x+6

Composite functions
ANSWERED Find the domain of the coposite function $$f\circ g$$

$$\displaystyle{f{{\left({x}\right)}}}=\frac{{5}}{{{x}+{9}}}$$
$$g(x)=x+6$$ 2020-11-25
The value of $$\displaystyle{f}\circ{g}$$ is:
$$\displaystyle{f{{\left({g{{\left({x}\right)}}}\right)}}}={f{{\left({x}+{6}\right)}}}$$
$$\displaystyle=\frac{{5}}{{{\left({x}+{6}\right)}+{9}}}$$
$$\displaystyle=\frac{{5}}{{{x}+{15}}}$$
therefore the composite function fog is:$$\displaystyle{f{{\left({g{{\left({x}\right)}}}\right)}}}=\frac{{5}}{{{x}+{15}}}$$
as we know that the domain of the function is the set of values of the independent variable for which the function is defined.
so we can notice that the composite function $$\displaystyle{f}\circ{g}$$ is defined when the denominator of $$\displaystyle{f}\circ{g}$$ is not equal to zero.
that implies
$$\displaystyle{x}+{15}\ne{0}$$
$$\displaystyle{x}\ne-{15}$$
therefore the composite function $$\displaystyle{f}\circ{g}$$ is defined for all values of x except x=-15.
therefore the domain of the composite function $$\displaystyle{f}\circ{g}$$ is $$\displaystyle:{\left\lbrace{x}{\mid}{x}\ne−{15}\right\rbrace}$$
therefore the second option is correct.