Question

Evaluate (Fofg)(x)

……..

Answer 1

First note the domains of f and g :

domain f(x) : x ≠ -16

domain g(x) : x ≠ 1, x ≠ -1

Then the domain of f ○ g is

15/(x ² - 1) ≠ -16

which can be "solved" for x :

`(15)/(x^2-1) = -16 \implies x^2-1 = -(15)/(16) \implies x^2 = \frac1{16} \implies x = \pm\frac14`

so that, in addition to domain of g, the domain of the composite function is

domain (f ○ g)(x) : x ≠ 1/4, x ≠ -1/4, x ≠ 1, x ≠ -1

The function itself can be evaluated as

`(f\circ g)(x) = f(g(x)) = f\left((15)/(x^2-1)\right) = ((15)/(x^2-1))/((15)/(x^2-1)+16)`

Simplifying a bit, we end up with

`(f\circ g)(x) = (15)/(15+16(x^2-1)) = (15)/(15+16x^2-16) = \boxed{(15)/(16x^2-1)}`