Identity Function Clarification: Definition & Examples

[Gear300]

From what I was reading, the apparent definition goes as: The Identity Function on E is the function I_E from E into E defined by I_E(x) = x. Since I_E is the set of all ordered pairs (x,x) such that x ϵ E, I_E is also called the diagonal subset of E x E.

If f is a function from E into F, clearly
1. f o I_E = f,
2. I_F o f = f, in which o is a composition operation

I understood 1., but I'm stuck on understanding on how 2. works. The definition is also confusing me; by how I read it, the identity function is the original function operating on itself...which, as stated, confuses me...any clarifications?

 

[Pere Callahan]

What is $(I_F\circ f)(x)$ for some element x of E?

 

[Gear300]

What is $(I_F\circ f)(x)$ for some element x of E?

oh...I sort of understand now...interesting...your presentation suddenly made sense to me...thanks.

In addition, could I get an example of an identity function for some arbitrary function (so I may further clarify my thoughts)?

 

[Pere Callahan]

The identity function does not depend on any arbitrary function. It simply is the function $I_E:E\to E$ which returns its argument unchanged, that is $I_E(x)=x$ for all x in E. For any set E, there is exactly one such function.

 

[Gear300]

I see...your answer clarifies things for me...thanks.