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 :

$\dfrac{15}{x^2-1} = -16 \implies x^2-1 = -\dfrac{15}{16} \implies x^2 = \dfrac1{16} \implies x = \pm\dfrac14$

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

$(f\circ g)(x) = f(g(x)) = f\left(\dfrac{15}{x^2-1}\right) = \dfrac{\dfrac{15}{x^2-1}}{\dfrac{15}{x^2-1}+16}$

Simplifying a bit, we end up with

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