Show that function composition is associative by showing that ((f ∘g) ∘h)(x)=(f ∘(g ∘h))(x).

Name: Problem 91, Chapter 9: Intermediate Algebra : Concepts and Applications
Uploaded: 2020-06-13T18:38:11-08:00
Duration: 5 min 33 s
Channel: Caleb Wood
Description: Show that function composition is associative by showing that ((f ∘g) ∘h)(x)=(f ∘(g ∘h))(x).

Show that function composition is associative by showing...

Key Concepts

Function Composition

Function composition is the process of applying one function to the result of another, effectively combining them into a single operation. This concept is fundamental in mathematics, as it allows the construction of more complex functions from simpler ones by ensuring that the output of one function becomes the input of another.

Associativity

Associativity is a property of an operation that indicates the manner in which operands are grouped does not affect the final result of the operation. In the context of function composition, this means that no matter how the functions are nested or grouped during composition—whether composing f with g first before applying h, or composing g with h first and then applying f—the outcome will be the same.

Transcript

00:01 This problem, we need to, our main goal is to determine whether or not the associate property exists for composition functions.

00:10 In other words, if i first composed f and g and then compose that function with h, does that equal the same expression as composing first g and h and then using that in the composition of f? so i'm going to write my three sample functions on the right -hand side.

00:34 I'm going to say that f -of -x is equal to 2x, g of x is equal to x squared, and h -of -x will simply be equal to x -minus 2.

00:56 Three relatively simple functions to use.

01:00 So let's compose each function is shown on both sides in both ways here with the associate property and see it to come out to be the same expression.

01:11 So first we have f composed of g of h.

01:20 F composed of g in that function will later be composed of h.

01:26 So with my color coding, let me rewrite this as in red, f composed of my green, g been composed of h in blue of x to equal f composed of my green g being first composed of h input of x all right so we need to find f composed of g now we'll equal f of g of x being x squared and then that will eventually be composed of h of x of x so this is simplified down to, in red, if you plug in x squared, f of x will get 2 times x squared.

03:01 But then this needs to be composed of h still of x.

03:10 Let's roll down a little bit.

03:13 So we need to plug in h of x into 2x squared, right? that will give us an expression of 2 times x minus 2 quantity squared...

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