Mathematics

Problem

Let

f(x)=\dfrac{x+1}{x-1},\qquad g(x)=\dfrac{1}{x^{2}-4},

with their natural domains in

\mathbb R

. a. (i) Determine a simplified algebraic expression for the composite

(g\circ f)(x)

. b. (ii) Find all real numbers

a

for which

(g\circ f)(a)

is not defined. Give clear justification for every value obtained. c. (iii) Evaluate

(g\circ f)(0)

and

(g\circ f)(3)

, stating whether each value is finite, zero, or undefined.