Question

Find the composite functions (f ∘ g) and (g ∘ f). what is the domain of each composite function? \[ \begin{array}{rcl} f(x) & = & \dfrac{{\color{red}3}}{x} \\ & & \\ g(x) & = & x^2 - {\color{red}81} \end{array} \]

Answer 1

Always remember two steps while finding the domain of composite function.

1) First find the domain of inside/ input function( A common mistake is to skip this point).

2) Find the domain of new function after performing the composition.

In our case, the given functions are

`f(x)=(3)/(x)` ,
`g(x)= x^(2) -81`

Now,

1) (f ° g)(x)

= f[g(x)]

=f[x²-81]

`= (3)/(x^(2)-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
`(3)/(x^(2)-81)`.

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

`=(9-81x^(2))/(x^(2))`

First find the domain of input function 3/x which is set of all real numbers except x=0

`(9-81x^(2))/(x^(2))`

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)