# Thread: question regarding the composition of functions

1. ## question regarding the composition of functions

Simple question. Suppose $f(g(x)) = g(f(x))$. Then must one of the following be true?

1. One of the functions is the identity.

2. The two functions are inverses.

2. ## Re: question regarding the composition of functions

Originally Posted by icemanfan
Simple question. Suppose $f(g(x)) = g(f(x))$. Then must one of the following be true?

1. One of the functions is the identity.

2. The two functions are inverses.
Consider the special case f(x) and g(x) are the same function.

CB

3. ## Re: question regarding the composition of functions

Suppose then the criteria become:

1. One of the functions is the identity

2. The two functions are inverses

3. The two functions are the same

Must one of these statements be true?

4. ## Re: question regarding the composition of functions

What if f(x)= ax, g(x)= bx, where a and b are constants?

5. ## Re: question regarding the composition of functions

f o g:x = g o f:x if one of the following statement holds

i) one of the functions is the identity
ii)the two function are inverses
iii) the two functions are the same
iv) the two functions are of the form f(x) = ax + b = x + b that is a =1
g(x) = cx + d = x + d that is b = 1

we call this functions translations

v) f and g have the same fixed point.
fixed point is the point X0 = (xo,xo) such that f(xo) = xo and g(x0) = xo

fixed point of f = -b/(a-1)