The document defines three functions f(x), g(x), and h(x) and asks to evaluate compositions of these functions. (a) It finds that f ◦ g ◦ g(x) = x4 - 2 and h ◦ g ◦ h(x) = 4x. (b) It determines that (f ◦ g ◦ h)(x) = 0 when x = 1/2. (c) It shows that (g ◦ g)(x) = x4, (g ◦ g ◦ g)(x) = x8, and more generally (g ◦ g

1. Exercice 29
Si f (x) = x − 2 g(x) = x 2 h(x) = 2x , alors :
(a) f ◦ g ◦ g = f (g(g(x)))
Comme g(g(x)) = g(x 2 ) = (x 2 )2 = x 4
⇒ f (x 4 ) = x 4 − 2
⇒ f ◦ g ◦ g = x4 − 2
h ◦ g ◦ h = h(g(h(x)))
Comme g(h(x)) = h(x)2 = (2x )2 = x 2x
2x )
⇒ h(22x ) = 2(2

2. Exercice 29 (suite..)
(b) (f ◦ g ◦ h)(x) = 0
Comme g(h(x)) = 22x
f (22x ) = 22x − 2 = 0
22x = 2
2x = 1
x = 1/2

3. Exercice 29 (suite..)
(c)
(g ◦ g)(x) = g(g(x)) = g(x 2 ) = (x 2 )2 = x 4
3
(g ◦ g ◦ g)(x) = g(x 4 ) = (x 4 )2 = x 8 = x 2
4
(g ◦ g ◦ g ◦ g)(x) = g(x 8 ) = (x 8 )2 = x 16 = x 2
n
g ◦ g ◦ g ◦ g · · · g = x2
n fois