Question

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

Composite functions
ANSWERED
asked 2020-11-24

Find the domain of the coposite function \(f\circ g\)

\(\displaystyle{f{{\left({x}\right)}}}=\frac{{5}}{{{x}+{9}}}\)
\(g(x)=x+6\)

Answers (1)

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.
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