Mathematics

Problem

Let the functions

f

and

g

be given by

f(x)=|x-3|,\qquad g(x)=x^{2}-4x+7,

where

x\in\mathbb R

. a. (i) Write

(g\circ f)(x)

explicitly, expressing your answer in a piecewise form without the modulus sign. b. (ii) Determine the range of

(g\circ f)(x)

. Show all necessary working. c. (iii) Find all real numbers

t

for which

(g\circ f)(t)=11

. Comment on how many such numbers exist.