Mathematics

Problem

The function

f:\mathbb R \to \mathbb R

is defined piece-wise by

f(x)=\begin{cases} x^{2}+2, & x\le -1,\\ 3x+1, & -1<x<2,\\ x^{2}-4x+5, & x\ge 2. \end{cases}

a. (a) On the same set of axes, sketch the graph of

y=f(x)

, marking clearly the points of transition at

x=-1

and

x=2

. b. (b) For each of the three pieces, state whether that portion of the graph represents a one–one correspondence between

x

and

y

. c. (c) Determine the largest subset

D\subseteq\mathbb R

such that the restricted function

f|_{D}

is one–one. d. (d) Find an explicit algebraic expression for

f^{-1}(x)

corresponding to the domain obtained in part (c), and state the domain and range of

f^{-1}