When are two functions equal? A function is a kind of relation, which means it's a kind of set. So two functions being equal means, fundamentally, that each is a subset of the other. That's inconvenient, and it doesn't really jibe with our understanding of functions as encoding *rules*.

**Theorem About When Functions Are Equal** (TAWFAE)**.** Let and be functions. Then if and only if:

- , and
- .

**Proof. **This theorem sort of looks like it will be complicated to prove, but let's keep our wits about us and rely on logic to help us out.

Globally, this is a biconditional proof so it has two directions:

() Assume . We need to prove two things:

(*clause 1*) This is the claim that two sets are equal, so we have to prove each is a subset of the other:

() Let . Then . Since , this means . So .

() Let . Then . Since , this means . So .

(*clause 2*) This is a universal claim, so we start the same way as always: Let . Then . We can rewrite this as . On the other hand, we know from clause 1 that , so . We can rewrite this as . But since , we actually have .

Since and , and is a function, .

So .

() Assume both clauses 1 and 2 hold. In this direction, we need to prove two sets are equal.

() Let . Then for some and some . , so by clause 1, . Then there's some with . We can rewrite in function notation as and . By clause 2, we know , so . That is, .

() This is so similar to , I'll let you work it out.

Since we proved both directions, we're done. .

This theorem gives us a new way to prove two functions are equal, should we ever need it: we establish clause 1 ("they have the same domain") and clause 2 ("they have the same rule").

### Restrictions and Extensions

In middle and high school algebra courses, we often ask questions like "What's the domain of ?". According to the TAWFAE, this is actually a dumb question: TAWFAE says in order to know what a function is, we need to know what the domain is. So a question like doesn't actually contain enough information to be answerable.

What the question is really asking, though is: *what's the most natural*, or maybe *what's the largest possible* domain on which this rule makes sense?

As a middle-school student, I was very confused by questions like this one:

Find the domain of .

My answer would usually go like this: that function is really the same as , the domain of which is all of . And I would always get that question wrong. Why?

The functions and do indeed have the same rule, in the sense that, whenever makes sense, we have

**But **the natural domain of is and the natural domain for is ; so that means the domains are different -- hence by the TAWFAE, we know . Nevertheless, the two functions *are* related.

**Definition.** If and , we define the *restriction* of to as

That is, has the same rule as , just we only care about inputs that come from .

Equipped with this notion, we can describe the relationship between and :

**Definition.** If , and , and there is with , we call an *extension *of .

Here's an illustration: if we let denote the function displayed in green, denote the function displayed in blue, and the function displayed in red, then we have that:

- is an extension of , from to all of
- is an extension of , from to