Let the functions and be given by f(x)=|x-3|

MATHEMATICS · Multi-part · A Level · Mathematics · Algebra and Functions · Algebra and Functions

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

. (i) Write

(g\circ f)(x)

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

(g\circ f)(x)

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

t

for which

(g\circ f)(t)=11

. Comment on how many such numbers exist.