Mathematics

Problem

Consider the functions

f(x)=x^{2}-4

(domain

\mathbb{R}

) and

g(x)=\operatorname{sgn}(x)=\begin{cases}1, & x>0\\0, & x=0\\-1, & x<0\end{cases}

. a. (i) Form the composite

(g\circ f)(x)

and express it as a piecewise–defined function without using the signum notation. b. (ii) Determine the domain of

(g\circ f)

and justify your answer. c. (iii) Find the range of

(g\circ f)

. d. (iv) Sketch the graph of

(g\circ f)

, clearly indicating all intercepts and points of discontinuity, if any.