Let f, g be two invertible functions from a set S to itself. Let h=f ∘g; that is, h is the compo

Let f, g be two invertible functions from a set S to itself....

Key Concepts

Order of Function Application and Inversion

When composing functions, the order of operations is significant; the function applied last is the first to be 'undone' when taking an inverse. This reversal of order underpins the result that the inverse of a composite function f ? g is given by g?¹ ? f?¹, reflecting the necessity to reverse both the sequence and the effect of the functions involved.

Identity Function

The identity function acts as a neutral element in function composition, meaning that when a function is composed with the identity function, the result remains unchanged. It plays a crucial role in verifying that a function and its inverse properly cancel each other, as f(f?¹(x)) and f?¹(f(x)) both yield the identity function, leading back to the original input.

Invertible Function

An invertible function is one for which every output is produced by exactly one input, allowing the definition of an inverse function that 'undoes' the effect of the original function. This inverse function, denoted as f?¹ for a function f, satisfies the conditions f(f?¹(x)) = x and f?¹(f(x)) = x, ensuring each application leads back to the original value.

Composition of Functions

Function composition is the process of applying one function to the results of another (written as f ? g), where the output of g becomes the input of f. Understanding composition is key because it demonstrates how complex transformations can be built from simpler ones, and establishes that the overall operation has structured behavior.

Transcript

00:01 I want to show that h has an inverse, right? you want to show that the function h has an inverse.

00:08 So you know that if a function is an inverse of another function.

00:13 For example, if f inverse, this one is the inverse of f of x, then f inverse of f of x must give you x.

00:21 If this relation exists, then it means that this is the inverse of f of x.

00:29 Okay, so you're going to use that relation.

00:35 So what we're going to do is take g inverse prime of f inverse, g inverse of f inverse of h of x.

00:45 We want to see that if this thing is going to give us an x finally at the end of the day, right? so that that will show that h has an inverse, right? so i want to do this one right here.

01:00 So when we do this, we know that.

01:03 H inverse, we know that this h of x is, this h of x is given as, you know, f of g of x, right? this h inverse is f of g of x as given.

01:22 So in place of this guy, i'm going to put this guy, right? so this is going to be, it's going to be g inverse of f inverse of f inverse of f of g of x.

01:38 Okay, so now, then this one is gonna be g inverse of f inverse of f of g of x.

01:51 There's a lot of offs, but that is what, so it is just this one being evaluated at this thing.

02:00 That's what is happening.

02:01 So you can see that this f inverse is gonna take away this f.

02:05 So what is left is just just g.

02:07 So you're going to have something like g inverse of g of x, right? because this f inverse is taking care of this f of x...

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