est
The basic compositions are the following:
f(x) = 1
g(x)
↓
x
-
1
=
C
Og
g(f(x))
f(f(x))
g(g())
1-
=
1
=
1
-
1
=
-
x
1
x) =
Same funtion
x
=
-
=
Down below are more complex compositions :
i flg(f(x))
=
1
-
1
x
-
-
=
1
1
-
x
1(
1
-
x)
-
x
-
↓
1
x
-
=
x
g(f(g(x)))
+
=
1
=
1
-
2)
-
1
·
=
1-
ii)
g(g(f()))
=
iii)
f(f(g(x)))
=
iv)
f(f(()) =
1
1
-
-
1
c)
-
2) =
.
[l-e-x
·
f (f()) = x
:
f(f(f()))
· f(fff (f(x)))) = 1
-
=
1
-
x
x
Similarly :
g(g) =: g(g(g()) =
Composition of length 4:
g(f(+g()))) g()
=
=
=
30
=
1
f(f(g(f(x))))
=
f()
=
1
-
()
=
=
=
=
grfg(f())))
x
1
=
↓
I
D
.
El-x
-
(g (g (f (c)))) = 1
gigfry(r)
-
-
=
2&
x
=
x
=I
( x) =
1
=
75
-
1
x
=
1
By making the function with the length of 4 we can tell that f(f(g(f(x)))) = g(f(x))
and g(f(g(f(x)))) = f(g(x)). The more compositions of these functions I make the
more I come across functions that I have already done.
Domain
f(x) = 1
xE1
-
-
,
x
exx 0
g(x) =
xf0
-
a
( *
g(f(x)) =
x
xt1
+xxa
,
g(g(f(x))) =
3 70
-
*
x
*
The largest domain is 0 and 1. All function are de ned at any value of x
besides 0 and 1.