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Function composition

  • Thread starter estro
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  • #1
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I have hard time understanding composition of functions in several variables.
Suppose [tex]f(x,y)=(xcosy,\frac {x-y}{y-x})[/tex], how should I write f(f(x,y))?
 

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  • #2
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Hi estro! :smile:

I have hard time understanding composition of functions in several variables.
Suppose [tex]f(x,y)=(xcosy,\frac {x-y}{y-x})[/tex], how should I write f(f(x,y,z))?
I assume you mean f(f(x,y))?

Well, we have

[tex]f(f(x,y))=f(xcosy,\frac {x-y}{y-x})[/tex]

Can you work this out further?
 
  • #3
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I have hard time understanding composition of functions in several variables.
Suppose [tex]f(x,y)=(xcosy,\frac {x-y}{y-x})[/tex], how should I write f(f(x,y,z))?
You can't. f maps R2 to R2, so f(x, y, z) has too many inputs. Did you mean f(f(x, y))?
 
  • #4
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Hi estro! :smile:



I assume you mean f(f(x,y))?

Well, we have

[tex]f(f(x,y))=f(xcosy,\frac {x-y}{y-x})[/tex]

Can you work this out further?
Thank you for the fast response, yes I mean f(f(x,y)). [fixed it in my first post]

So if i get it right: [tex]f(f(x,y))=((xcosy)cosy,\frac {x-\frac {x-y} {y-x}} {\frac {x-y} {y-x}-x})[/tex]

Please give me feedback...=)
 
  • #5
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3,281
Thank you for the fast response, yes I mean f(f(x,y)). [fixed it in my first post]

So if i get it right: [tex]f(f(x,y))=((xcosy)cosy,\frac {x-\frac {x-y} {y-x}} {\frac {x-y} {y-x}-x})[/tex]

Please give me feedback...=)
Hmm, let me explain it differently:

[tex]f(f(x,y))=f(x\cos{y},\frac{x-y}{y-x})=(u\cos{v},\frac{u-v}{v-u})[/tex]

with

[tex]u=x\cos{y}~~\text{and}~~v=\frac{x-y}{y-x}[/tex]

Can you do it now?
 
  • #6
I like Serena
Homework Helper
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Hi estro. :)

I have hard time understanding composition of functions in several variables.
Suppose [tex]f(x,y)=(xcosy,\frac {x-y}{y-x})[/tex], how should I write f(f(x,y))?
Usually, when we have these confusing expressions, it pays to introduce helper variables.

Let's define:
[tex]u = x \cos y[/tex]
[tex]v = \frac {x-y}{y-x}[/tex]

Then
[tex]f(f(x,y)) = f(u,v) = (u \cos v,\frac {u-v}{v-u})[/tex]

Now substitute u and v ...


EDIT: Wow, micromass just posted exactly what I just posted at the same time! :bugeye:
 
Last edited:
  • #7
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Thank you all guys, I think now I get it:

[tex]f(f(x,y))=(xcosycos(\frac {x-y}{y-z}), \frac {xcosy-\frac {x-y} {y-x}} {\frac {x-y}{y-x}-xcosy})[/tex]

It seems to me that this composition is violation of human rights...=)
 
  • #8
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3,281
Thank you all guys, I think now I get it:

[tex]f(f(x,y))=(xcosycos(\frac {x-y}{y-z}), \frac {xcosy-\frac {x-y} {y-x}} {\frac {x-y}{y-x}-xcosy})[/tex]

It seems to me that this composition is violation of human rights...=)
Seems correct! :smile:
I'll warn the Geneva convention :biggrin:
 

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