ANSWER TO PART A
The mapping for the reflection in the line [tex]y=x[/tex], is given by
[tex](x,y)\rightarrow (y,x)[/tex].
That is the coordinates swap position .
The only way we can construct a function [tex]f(x)[/tex], such that;
[tex](x, f(x))\rightarrow (f(x),x)[/tex] are equal is when
[tex]f(x)=x[/tex].
So that when [tex]x=a, f(a)=a[/tex] .
The mapping then becomes
[tex](a,a)\rightarrow (a,a)[/tex].
Therefore the function, [tex]f(x)=x[/tex] is the function whose reflection in the line
[tex]y=x[/tex] is itself.
ANSWER TO PART B
The function is symmetrical with respect to the origin. That is to say the function is an odd function.
A function is symmetric with respect to the origin, if it satisfies the condition,
[tex]f(-x)=-f(x)[/tex]
For instance,
[tex]f(a)=a[/tex]
[tex]f(-a)=-a[/tex]
Since
[tex]f(-a)=-a=-f(a)[/tex]
We say the function is symmetric with respect to the origin.
