# Function composition

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))?

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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
Mentor
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))?

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

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

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
Homework Helper
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! Last edited:
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...=)

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 