Falak Kinney

2020-11-24

Find the domain of the coposite function $f\circ g$
$f\left(x\right)=\frac{5}{x+9}$
$g\left(x\right)=x+6$

faldduE

The value of $f\circ g$ is:
$f\left(g\left(x\right)\right)=f\left(x+6\right)$
$=\frac{5}{\left(x+6\right)+9}$
$=\frac{5}{x+15}$
therefore the composite function fog is:$f\left(g\left(x\right)\right)=\frac{5}{x+15}$
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 $f\circ g$ is defined when the denominator of $f\circ g$ is not equal to zero.
that implies
$x+15\ne 0$
$x\ne -15$
therefore the composite function $f\circ g$ is defined for all values of x except x=-15.
therefore the domain of the composite function $f\circ g$ is $:\left\{x\mid x\ne -15\right\}$
therefore the second option is correct.

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