ENSEMBLES ET APPLICATIONS exercices

I f I
f
f
f I =R
f
f
f
fR
∀x∈R,∃y∈R|y=f(x)
∃y∈R| ∀x∈R, y =f(x)
∀y∈R,∃x∈R|y=f(x)
∃x∈R| ∀y∈R, y =f(x)
∀x∈R,∃M∈R|f(x)≤M
∃M∈R| ∀x∈R, f(x)≤M
f I
x0∈I
∀ε > 0,∃η > 0| ∀x∈I, (|x−x0|< η =⇒ |f(x)−f(x0)|< ε).
f x0
n∈N
n2n
E={a, b, c}
a∈E
a⊂E{a} ⊂ E ∅ ∈ E∅ ⊂ E
√x2−2x+ 2 = x−2x
A B E A ∪B=A∩B
A=B

A B C E
A\(B∩C)=(A\B)∪(A\C)
R=[
n∈N
[−n, n]{0}=\
n∈N∗−1
n,1 n
fRf(x) = 2x
1 + x2
x x0f(x) = f(x0) (x−x0)(1 −xx0) = 0
f
y∈R
y yx2−2x+y= 0
x∈R
f
Im(f)=[−1,1]
f f(x) = 2x+ 1
3x+ 2 R\−2
3R\2 3
f:C−→ Cf(z) = z2
f
P={z∈C; Re(z)>0}
g f P
gPC
E F Rf E F f(x) = x4
f
E=RF=RE=R+F=RE=RF=R+E=R+F=R+
E=F=R
f(R)f(R+)f([1,2]) f([−1,2]) f−1(R+)f−1(R∗
+)f−1([1,4]) f−1([−1,4]) f−1({y})
y∈R
fC\ {1}f(z) = z+ 1
z−1
f−1(R)f−1(iR)f−1(U)
f(U)

f: [0,2[∪]2,+∞[−→ ]− ∞,0]∪]1,+∞[
x7−→ x2
x2−4
f:R2−→ R2
(x, y)7−→ (x+y, x −y)
f
g:R2−→ R2
(x, y)7−→ (x+y, x +y)
f:N−→ Z
f(n) = 
n
2n
−n+ 1
2
f
f:N−→ N
n7−→ 2n
f
g:N−→ N
g(x) = x
2x
x x
g◦f f ◦g
a b c c 6= 0 a2+bc 6= 0
fR\na
cof(x) = ax +b cx −a
Im(f)⊂R\na
co
x∈R\na
cof(f(x))
f

E F G f E F g F G
g◦f g
g◦f f
E F G f E F g F G
g◦f g f
g◦f f g
E F f ∈ A(E, F )g∈ A(F, E)f◦g◦f
f g
p E E p ◦p=p
p p = IdE
E f E E f ◦f◦f=f
f f
E f g E E
f◦g◦f=g g ◦f◦g=f f
g
y∈E x ∈E f(y)=(f◦g◦f)(x)
x0∈E y =g(x0)
g
f g
E F f E F A ⊂E B ⊂F
f(A∩f−1(B)) = f(A)∩B