Always remember two steps while finding the domain of composite function.
1) <em>First find the domain of inside/ input function( A common mistake is to skip this point).</em>
2) <em>Find the domain of new function after performing the composition.</em>
In our case, the given functions are
,
Now,
1) (f ° g)(x)
= f[g(x)]
=f[x²-81]
(this is (f ° g)(x) function)
Now, its domain
First find the domain of input function which is x²-81, its domain is the set of all real numbers.
domain of new function after performing composition which is .
x²-81=0 ⇒ (x-9)(x+9)=0
(x-9)=0 , (x+9)=0
x=9 , x=-9 (exclude these points from the domain because anything/0 does not exist in math)
So, the Domain of Composite function (f °g)(x) is the set of all real numbers except x=9, x=-9.
Domain= (-infinity, -9)U(-9, 9)U(9, infinity)
2) (g °f)(x)
= g[f(x)]
= g[3/x]
= (3/x)² -81
= 9/x² -81
First find the domain of input function 3/x which is set of all real numbers except x=0
Domain of above function after performing composition is set of all real numbers except x=0
So, the domain of (g° f)(x) is the set of all real numbers except x=0.
Domain= (-infinity, 0)U(0, infinity)