# Finding the Domain of a Composite Function. Find the domain of f circ g for the functions. f(x)=x^2-9 and g(x)=sqrt(9)-x^2

Finding the Domain of a Composite Function. Find the domain of $f\circ g$ for the functions.
$f\left(x\right)={x}^{2}-9\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}g\left(x\right)=\sqrt{9}-{x}^{2}$
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Solution:
The functions are $f\left(x\right)={x}^{2}-9\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}g\left(x\right)=\sqrt{9}-{x}^{2}$
Now the composite function is obtained as,
$f\circ g=f\left(g\right)$
$=f\left(\sqrt{9-{x}^{2}}\right)$
${\left(\sqrt{9-{x}^{2}}\right)}^{2}-{9}^{d}$
$=9-{x}^{2}-9$
$=-{x}^{2}$
So the composite function is defined for all value of x.
Hence, the domain for the composite function is real number.