Consider the function given by .

MATHEMATICS · Multi-part · AoPS · Mathematics · Special Functions · Special Functions

Problem

Consider the function

h: \mathbb R\to\mathbb R

given by

h(x)=\sqrt{x^{2}}

. (i) Rewrite

h(x)

without the radical sign and prove that the two expressions represent the same function on

\mathbb R

. (ii) Name the parent family of

h

and list two distinctive graphical features that support this classification. (iii) Prove that

h

is an even function and describe how this property is reflected in its graph. (iv) Compare

h

with the identity function at

x=-3,0,3

and comment on the relative positions of their graphs at these points.