Mathematics

Problem

Consider the function

s: \mathbb R\to\mathbb R

given by

s(x)=x^{2}+1

. A student claims that

g(y)=\sqrt{y-1}

is the inverse of

s

. a. (a) Show, with reasons, whether

s

is injective and whether it is surjective onto

\mathbb R

. b. (b) Evaluate the compositions

g\bigl(s(x)\bigr)

and

s\bigl(g(y)\bigr)

and specify the set of real numbers for which each composition is defined. c. (c) Using the results of part (b), explain precisely why

g

cannot serve as an inverse to

s

on the stated domain and co-domain.