Function Composition: Understanding Multi-Var.

[estro]

I have hard time understanding composition of functions in several variables.
Suppose $$f(x,y)=(xcosy,\frac {x-y}{y-x})$$, how should I write f(f(x,y))?

 

[micromass]

Hi estro!

I have hard time understanding composition of functions in several variables.
Suppose $$f(x,y)=(xcosy,\frac {x-y}{y-x})$$, how should I write f(f(x,y,z))?

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

Well, we have

$$f(f(x,y))=f(xcosy,\frac {x-y}{y-x})$$

Can you work this out further?

 

[Mark44]

I have hard time understanding composition of functions in several variables.
Suppose $$f(x,y)=(xcosy,\frac {x-y}{y-x})$$, 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))?

 

[estro]

Hi estro!



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

Well, we have

$$f(f(x,y))=f(xcosy,\frac {x-y}{y-x})$$

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: $$f(f(x,y))=((xcosy)cosy,\frac {x-\frac {x-y} {y-x}} {\frac {x-y} {y-x}-x})$$

Please give me feedback...=)

 

[micromass]

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: $$f(f(x,y))=((xcosy)cosy,\frac {x-\frac {x-y} {y-x}} {\frac {x-y} {y-x}-x})$$

Please give me feedback...=)

Hmm, let me explain it differently:

$$f(f(x,y))=f(x\cos{y},\frac{x-y}{y-x})=(u\cos{v},\frac{u-v}{v-u})$$

with

$$u=x\cos{y}~~\text{and}~~v=\frac{x-y}{y-x}$$

Can you do it now?

 

[I like Serena]

Hi estro. :)

I have hard time understanding composition of functions in several variables.
Suppose $$f(x,y)=(xcosy,\frac {x-y}{y-x})$$, how should I write f(f(x,y))?

Usually, when we have these confusing expressions, it pays to introduce helper variables.

Let's define:
$$u = x \cos y$$
$$v = \frac {x-y}{y-x}$$

Then
$$f(f(x,y)) = f(u,v) = (u \cos v,\frac {u-v}{v-u})$$

Now substitute u and v ...EDIT: Wow, micromass just posted exactly what I just posted at the same time!

 

[estro]

Thank you all guys, I think now I get it:

$$f(f(x,y))=(xcosycos(\frac {x-y}{y-z}), \frac {xcosy-\frac {x-y} {y-x}} {\frac {x-y}{y-x}-xcosy})$$

It seems to me that this composition is violation of human rights...=)

 

[micromass]

Thank you all guys, I think now I get it:

$$f(f(x,y))=(xcosycos(\frac {x-y}{y-z}), \frac {xcosy-\frac {x-y} {y-x}} {\frac {x-y}{y-x}-xcosy})$$

It seems to me that this composition is violation of human rights...=)

Seems correct!
I'll warn the Geneva convention