Inverse Functions and Composition

Review Results about Inverse Functions (From Section 7.2 and My Lectures)

Definition of Inverse Function. (My definition. Not how the book defines it.)

To say that functions f and g are inverses of one another means that

The functions have reversed domains & codomains. That is, f : A → B and g: B → A
Their behavior is related by: g(b) = a ⇔ f (a) = b.

Theorem 7.2.2 and Theorem 7.2.3 (Combined and reworded):

If a function f is both one-to-one and onto,
then f has an inverse function g, and g is also both one-to-one and onto.

"Missing" Theorem A (Not in the book, but very useful.)

If a function f has an inverse function g
then both f and g are one-to-one and onto.

Results about Inverse Functions and Function Composition (From Section 7.3 and My Lectures)

Theorem 7.3.2 About Composition of a Function with its Inverse

Suppose f : A → B and g : B → A.
If f ,g are inverses, then g ○ f = I_A and f ○ g = I_B.

"Missing" Theorem B (Not presented as a theorem in the book, but related to Sugg Exercise 7.3 # 25 and very useful. Notice that it is the converse of Theorem 7.3.2.)

Suppose f : A → B and g : B → A.
If g ○ f = I_A and f ○ g = I_B, then f ,g are inverses.

Results about One-to-One, Onto and Function Composition (From Section 7.3 and My Lectures)

Theorem 7.3.3 About Composition of two one-to-one functions

Suppose f : A → B and g : B → C.
If f ,g are both one-to-one, then g ○ f : A → C is also one-to-one.

Theorem 7.3.4 About Composition of two onto functions

Suppose f : A → B and g : B → C.
If f ,g are both onto, then g ○ f : A → C is also onto.

What about the Converses?

We saw in class that it is possible for g ○ f : A → C to be one-to-one and yet have g not be one-to-one. So the converse of the statement of Thereom 7.3.3 is not true, and is not a thoerem.

And we saw in class that it is possible for g ○ f : A → C to be onto and yet have f not be onto. So the converse of the statement of Thereom 7.3.4 is also not true, and is not a thoerem.

The following two statements are the all that can be said about the cases where g ○ f is known to be one-to-one or onto. These statements are not presented as theorems in the book. They appear as exercises in Section 7.3. (They are on your suggested exercise list.) Since they will be useful in the future, I state them as theorems here. (You will prove them in your homework.)

Theorem About the Situation where a Composition of Functions is Known to be One-to-One

Suppose f : A → B and g : B → C.
If g ○ f : A → C is one-to-one, then f is one-to-one.

Theorem About the Situation where a Composition of Functions is Known to be Onto

Suppose f : A → B and g : B → C.
If g ○ f : A → C is onto, then g is onto.