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

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

,

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)

step-by-step explanation:

equation

you are asked to write the equation

  expression = 0

using the given expression. when you do that, you have

  2x^2-2x-12 = 0

factoring

you can see that each of the coefficients is even, so a factor of 2 can be removed immediately.

  2(x^2 -x -6) = 0

if you multiply the binomials (x+a)(x+b), you get

  (x +a)(x +b) = x^2 +(a+b)x +ab

this tells you the coefficient of x is the sum of two factors of the constant term. here, that means -1 is the sum of two factors of -6. we want to know those factors so we can factor the trinomial.

  -6 = -1×6 = -2×3 = -3×2 = -6×1

the factor pair that sums to -1 is {-3, 2}, so our factorization is

  2(x -3)(x +2) = 0

solutions

the solutions to this equation are the values of x that make the product zero. a product is zero only when only of the factors of the product is zero. here, that means

  x - 3 = 0

  x = 3 . . . add 3 to both sides of the equation

or

  x +2 = 0

  x = -2 . . . add -2 to both sides of the equation

the solutions to the equation are x=3 or x=-2. both of these values of x satisfy the equation—they make it be true.