Proving the graph of a composite function

In summary, the conversation discusses functions and graphs, specifically the sets Γf, Γg, Γ f ×C, and A×Γg. The goal is to show that Γ is the graph of g◦f, using the projection θ((a,b,c))=(a,c). The speaker is unsure how to proceed and asks for help.
  • #1
The1TL
26
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Homework Statement



Let f : A → B, g : B → C be functions where A,B,C are sets. ConsiderΓf ⊂A×B,the graph of f,Γg ⊂B×C,the graph of g. Now consider the sets Γ f ×C ⊂ A×B×C and A×Γg ⊂A×B×C. LetΓ=θ(Γf ×C∩A×Γg)⊂A×C where θ : A×B×C → A×C is the projection defined as θ((a,b,c))=(a,c). Show that Γ is the graph of g◦f.

Homework Equations





The Attempt at a Solution



I think that I should find a way to show that the first projection of gamma is a bijection. But i don't know how I can do that.
 
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  • #2
please help I've been trying for so long
 
  • #3
Then show us what you have been doing all that time!
 

FAQ: Proving the graph of a composite function

1. How do you prove that a given graph is a composite function?

To prove that a graph is a composite function, you need to show that it is made up of two or more individual functions. This can be done by breaking down the graph into its component functions and showing how they are connected to each other.

2. What is the process for proving the graph of a composite function?

The process for proving the graph of a composite function involves identifying the different functions that make up the composite function, determining their individual domains and ranges, and then combining them to create the composite function's overall domain and range. This can be done algebraically or by graphing the individual functions and then overlaying them to create the composite function's graph.

3. Can a composite function have more than two component functions?

Yes, a composite function can have any number of component functions. The only requirement is that the output of one function must serve as the input for the next function in the composition.

4. How do you know if a given graph is a composite function or a single function?

A graph is a composite function if it is made up of two or more individual functions connected to each other. If the graph is a continuous line with no breaks or disjointed parts, it is likely a single function. However, if there are multiple segments or distinct parts to the graph, it is likely a composite function.

5. Are there any special techniques or strategies for proving the graph of a composite function?

One helpful technique for proving the graph of a composite function is to use the composition of functions rule, which states that the composite of two functions is equal to the first function evaluated at the output of the second function. Additionally, it can be useful to graph the individual functions first and then combine them to create the composite function's graph.

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