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4
3
2
1
Outside the high school program
/ Homographic functions
17 exercises corrected
a
Homographic functions
(6 exercices)
Homographic and affine functions
(3 exercices)
Homographic functions: problems
(3 exercices)
Studies of other functions
(3 exercices)
1.
Homo
gr
aphic
functions
E.407
Consider
the
function
f
defined
b
y
:
f
:
x
↦−
→
3
x
+
1
x
−
3
1
Sho
w
that,
for
an
y
x
∈
R
\
3
,
w
e
ha
v
e
:
3
x
+
1
x
−
3
=
3
+
10
x
−
3
2
Determine
the
direction
of
v
ariation
of
the
function
f
on
the
in
terv
al
−∞
;
3
.
E.424
Consider
the
function
:
f
:
x
↦−
→
3
x
+
1
2
x
−
1
1
a
Establish
the
follo
wing
equalit
y
:
3
x
+
1
2
x
−
1
=
5
4
x
−
2
+
3
2
b
Establish
the
deca
y
of
the
function
f
on
the
in
terv
al
−∞
;
1
2
.
2
The
represen
tativ
e
curv
e
of
the
function
f
is
plotted
in
the
orthonormal
frame
b
elo
w
:
a
Graphically
,
determine
an
appro
ximate
v
alue
of
the
an
teceden
t
of
the
n
um
b
er
3
b
y
the
function
f
.
b
Algebraically
,
find
the
an
teceden
t
of
the
n
um
b
er
3
b
y
the
function
f
.
c
Graphically
,
determine
the
solutions
of
the
inequation
f
(
x
)
3
E.427
Consider
the
function
f
defined
b
y
the
relation
:
f
:
x
↦−
→
2
−
6
x
1
−
2
x
.
1
Giv
e
the
definition
set
of
the
function
f
.
2
Determine
the
v
alue
of
the
reals
a
and
b
v
erifying
:
2
−
6
x
1
−
2
x
=
a
1
−
2
x
+
b
3
Deduce
that
the
function
f
is
decreasing
on
1
2
;
+
∞
E.429
Consider
the
function
f
defined
b
y
:
f
:
x
↦−
→
2
x
−
1
x
+
1
1
Find
the
domain
of
the
function
f
2
Pro
v
e
the
follo
wing
equalit
y
:
2
x
−
1
x
+1
=
2+
−
3
x
+1
3
Sho
w
the
gro
wth
of
the
function
f
on
]
−∞
;
−
1[
and
also
on
]
−
1
;
+
∞
[
4
Sho
w
algebraically
that
2
has
no
an
teceden
t
under
the
function
x
.
E.4868
Consider
the
function
f
defined
b
y
:
f
(
x
)
=
6
·
x
−
7
3
−
2
·
x
1
Establish
the
follo
wing
equalit
y
:
f
(
x
)
=
−
3
+
2
3
−
2
·
x
2
Establish
the
direction
of
v
ariation
of
the
function
f
on
the
in
terv
al
−∞
;
3
2
.
3
Determine
the
set
of
an
teceden
ts
of
1
b
y
the
function
f
.
4
Solv
e
the
equation
:
f
(
x
)
=
2
·
x
E.419
Consider
the
function
f
defined
b
y
:
f
:
x
↦−
→
x
+
1
x
−
2
1
a
Giv
e
the
definition
set
of
the
function
f
.
b
Determine
the
image
of
3
b
y
the
function
f
.
c
Determine
the
an
teceden
ts,
for
the
function
f
,
of
the
n
um
b
ers
−
1
and
0
.
d
Justify
that
1
admits
no
an
teceden
t
b
y
the
function
f
.
2
Establish
for
an
y
x
∈
R
\
2
,
the
follo
wing
equalit
y
:
x
+
1
x
−
2
=
3
x
−
2
+
1
2.
Homo
gr
aphic
and
affine
functions
E.6053
Consider
the
function
f
defined
on
R
\{−
1
}
b
y
the
relation
:
f
(
x
)
=
x
−
3
x
+
1
In
the
reference
frame
O
;
I
;
J
giv
en
b
elo
w,
w
e
ha
v
e
plotted
the
curv
e
C
f
represen
tativ
e
of
the
function
f
.
h
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chapExoCorrec/424
sacados/424
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O
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f
chapExoCorrec/427
sacados/427
chapExoCorrec/429
sacados/429
chapExoCorrec/4868
sacados/4868
chapExoCorrec/419
sacados/419
chapExoCorrec/6053
sacados/6053
1
a
Determine
the
co
ordinates
of
the
t
w
o
p
oin
ts
A
and
B
on
the
curv
e
C
f
with
abscissa
1
and
3
resp
ectiv
ely
.
b
Determine
the
reduced
equation
of
the
line
(
AB
)
.
c
T
racer,
dans
le
rep
ère,
la
droite
(
AB
)
.
2
Consider
the
straigh
t
line
(Δ)
with
reduced
equation
:
(Δ):
y
=
3
x
−
4
a
Establish
the
follo
wing
factorization
:
−
3
x
2
+
2
x
+
1
=
(1
−
x
)(3
x
+
1)
b
Solv
e
algebraically
the
inequation
:
f
(
x
)
3
x
−
4
c
Deduce
the
relativ
e
p
osition
of
the
curv
e
C
f
relativ
e
to
the
straigh
t
line
(Δ)
.
d
Dra
w
the
straigh
t
line
(Δ)
in
the
reference
frame.
E.6054
Consider
the
function
f
defined
on
R
\{−
1
}
b
y
the
relation
:
f
(
x
)
=
3
x
+
1
x
+
1
In
the
reference
frame
O
;
I
;
J
giv
en
b
elo
w,
w
e
ha
v
e
plotted
the
curv
e
C
f
represen
tativ
e
of
the
function
f
.
1
Determine
the
reduced
equation
of
the
line
(
d
)
passing
through
the
t
w
o
p
oin
ts
of
the
curv
e
C
f
ha
ving
resp
ec-
tiv
ely
abscissae
−
2
3
and
1
.
2
Consider
the
straigh
t
line
(Δ)
with
reduced
equation
:
(Δ):
y
=
x
+
1
3
a
Establish
the
follo
wing
factorization
:
−
3
x
2
+
5
x
+
2
=
(2
−
x
)(3
x
+
1)
b
Solv
e
algebraically
the
inequation
:
f
(
x
)
x
+
1
3
c
Deduce
the
relativ
e
p
osition
of
the
curv
e
C
f
relativ
e
to
the
straigh
t
line
(Δ)
on
R
.
E.6052
Consider
the
function
f
defined
on
R
\{−
1
}
b
y
the
relation
:
f
(
x
)
=
x
−
3
x
+
1
In
the
reference
frame
O
;
I
;
J
giv
en
b
elo
w,
w
e
ha
v
e
plotted
the
curv
e
C
f
represen
tativ
e
of
the
function
f
.
1
a
Determine
the
co
ordinates
of
the
t
w
o
p
oin
ts
A
and
B
on
the
curv
e
C
f
with
abscissa
−
2
and
3
resp
ectiv
ely
.
b
Determine
the
reduced
equation
of
the
line
(
AB
)
.
c
T
racer,
dans
le
rep
ère,
la
droite
(
AB
)
.
2
Consider
the
straigh
t
line
(Δ)
with
reduced
equation
:
(Δ):
y
=
−
2
x
−
3
a
Algebraically
determine
the
solutions
of
the
equation
:
f
(
x
)
−
2
x
−
3
b
Deduce
the
relativ
e
p
osition
of
the
curv
e
C
f
relativ
e
to
the
straigh
t
line
(Δ)
.
c
Dra
w
the
straigh
t
line
(Δ)
in
the
reference
frame.
3.
Homo
gr
aphic
functions
:
pr
oblems
h
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O
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chapExoCorrec/6054
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x
x
y
y
C
f
(
d
chapExoCorrec/6052
sacados/6052
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4
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-2
2
4
6
J
O
C
f
E.2987
A
motorist
tra
v
els
on
departmen
tal
roads
a
distance
of
48
k
m
at
an
a
v
erage
sp
eed
of
64
k
m
=
h
,
then
tra
v-
els
the
remaining
x
k
m
on
the
freew
a
y
at
an
a
v
erage
sp
eed
of
115
k
m
=
h
.
1
Determine
the
duration
of
y
our
journey
on
departmen
tal
roads.
2
a
Justify
that
the
total
tra
v
el
time
w
as
:
t
=
3
4
+
x
115
b
Justify
that
the
a
v
erage
sp
eed
v
of
the
motorist,
o
v
er
the
en
tire
route,
is
written
as
a
function
of
x
:
v
=
460
x
+
22080
4
x
+
345
3
The
driv
er
kno
ws
that
he
has
co
v
ered
the
en
tire
route
at
an
a
v
erage
sp
eed
of
92
k
m
=
h
.
Determine
the
distance
tra
v
elled
on
the
freew
a
y
b
y
the
driv
er.
Then
giv
e
the
total
distance
of
his
journey
.
E.2989
In
a
plane
with
an
orthonormal
co
ordinate
system
(
O
;
I
;
J
)
,
consider
the
p
oin
t
A
with
co
or-
dinate
(
2
;
1
)
.
F
or
a
a
real
n
um
b
er
b
elonging
to
the
in
terv
al
2
;
+
∞
,
con-
sider
the
p
oin
t
M
with
co
ordinates
(
a
;
0
)
.
Note
N
the
p
oin
t
of
in
tersection
of
the
line
(
M
A
)
with
the
y-axis
and
note
b
the
ordinate
of
the
p
oin
t
N
.
W
e
note
P
the
p
oin
t
with
co
ordinates
(
a
;
b
)
.
The
graph
b
elo
w
represen
ts
these
differen
t
p
oin
ts
in
the
case
a
=
4
:
The
purp
ose
of
this
exercise
is
to
kno
w
the
p
osition
of
the
p
oin
t
P
as
a
function
of
the
v
alue
a
;
w
e
sp
eak
of
the
ˇ
lo
cus
géométrique
ı
of
the
p
oin
t
P
when
the
n
um
b
er
a
describ
es
the
in
terv
al
2
;
+
∞
.
P
art
A
:
1
Justify
that
the
p
oin
t
N
cannot
b
e
defined
for
a
=2
.
2
W
e’re
in
terested
in
the
p
osition
of
p
oin
t
P
when
a
=3
:
a
Place
the
p
oin
t
M
3
(
3
;
0
)
.
b
Place
the
p
oin
t
N
3
as
indicated
in
the
statemen
t.
What
is
the
ordinate
of
the
p
oin
t
N
3
?
c
Place
the
p
oin
t
P
3
as
sp
ecified
in
the
statemen
t.
3
Carry
out
the
same
pro
cedure
as
for
the
previous
ques-
tion
for
the
differen
t
v
alues
of
a
:
a
=
2.5
;
a
=
5
;
a
=
7
4
What
conjecture
can
b
e
made
ab
out
the
curv
e
formed
b
y
the
set
of
p
oin
ts
P
when
the
n
um
b
er
a
describ
es
the
in
terv
al
2
;
+
∞
?
P
art
B
:
1
Justify
that
the
directrix
of
the
line
(
M
N
)
is
:
1
2
−
a
2
Determine
the
reduced
equation
of
the
line
(
M
N
)
3
Determine
the
expression
of
b
as
a
function
of
a
.
4
Express
the
co
ordinates
of
p
oin
t
P
as
a
function
of
a
.
Confirm
the
conjecture
made
in
question
A
.
E.2990
4.
Studies
of
other
functions
E.3002
Consider
the
function
f
defined
b
y
the
relation
:
f
(
x
)
=
1
x
−
3
−
1
x
+
1
1
Giv
e
the
definition
set
of
the
function
f
.
2
Establish
the
follo
wing
equalities
:
f
(
x
)
=
4
(
x
−
3)(
x
+
1)
=
4
(
x
−
1)
2
−
4
3
a
Justify
that,
for
x
b
elonging
to
−∞
;
−
1
,
the
image
of
x
is
p
ositiv
e.
b
Establish
that
the
function
f
is
increasing
on
−
∞
;
−
1
.
4
In
a
frame
of
reference
O
;
I
;
J
orthonormal,
note
C
f
the
represen
tativ
e
curv
e
of
the
function
f
:
a
F
or
an
y
n
um
b
er
h
∈
R
suc
h
that
the
t
w
o
n
um
b
ers
1+
h
and
1
−
h
b
elong
to
D
f
,
v
erify
the
follo
wing
equalit
y
:
f
(1
−
h
)
=
f
(1+
h
)
b
Displa
ying
the
curv
e
C
f
on
y
our
calculator,
what
geo-
metric
prop
ert
y
do
es
the
curv
e
C
f
seem
to
p
ossess?
h
ttps:/
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y
M
I
N
x
y
M
I
N
x
y
M
S
I
N
chapExoCorrec/3002
sacados/3002
E.3025
1
Consider
the
function
f
whose
image
of
a
n
um
b
er
x
is
giv
en
b
y
the
relation
:
f
(
x
)
=
x
+
2
(2
x
−
1)(3
−
x
)
a
Determine
the
definition
set
of
the
function
f
b
Establish
the
follo
wing
equalit
y
:
f
(
x
)
=
1
2
x
−
1
+
1
3
−
x
c
Sho
w
that
for
an
y
x
∈
−∞
;
−
2
,
the
image
of
x
b
y
f
is
a
p
ositiv
e
n
um
b
er.
2
Let
g
b
e
the
function
defined
b
y
:
g
:
x
↦−
→
4
4
x
−
2
−
2
2
x
+
5
a
Determine
the
definition
set
of
the
function
g
.
b
Determine
the
v
alues
of
a
and
b
v
erifying
the
follo
wing
relationship
:
g
(
x
)
=
12
4
·
(
x
−
a
)
2
+
b
c
Establish
the
direction
of
v
ariation
of
the
function
g
on
the
in
terv
al
1
2
;
+
∞
E.440
1
Establish
the
equalities
:
a
1
+
2
x
+
1
=
x
+
3
x
+
1
p
our
x
=
−
1
.
b
3
x
2
−
7
x
+
6
x
−
1
=
3
x
−
4
+
2
x
−
1
p
our
x
=
1
.
c
1
x
+
1
+
1
x
−
1
=
2
x
x
2
−
1
p
our
x
=
1
and
x
=
−
1
.
2
Using
the
results
of
the
previous
question,
establish
the
follo
wing
statemen
ts
:
a
The
function
f
defined
b
y
x
↦−
→
x
+3
x
+1
is
decreasing
on
−
1
;
+
∞
.
b
The
function
g
defined
b
y
x
↦−
→
2
x
x
2
−
1
is
increasing
on
1
;
+
∞
.
5.
Unclassifie
d
financial
ye
ars
E.7333
Consider
the
function
f
defined
on
R
\{
1
}
b
y
the
relation
:
f
(
x
)
=
3
·
x
+
4
x
−
1
1
Establish
iden
tit
y
:
f
(
x
)
=
3+
7
x
−
1
2
Establish
that
the
function
f
is
decreasing
on
the
in
terv
al
1
;
+
∞
.
E.4517
Consider
the
homographic
function
f
defined
b
y
:
f
(
x
)
=
x
−
2
4
x
+
2
Consider
the
plane
equipp
ed
with
the
co
ordinate
system
(
O
;
I
;
J
)
orthonormal
represen
ted
b
elo
w
:
1
Justify
that
the
function
f
is
defined
on
R
\
−
1
2
:
2
Using
a
calculator
:
a
Dra
w
up
a
table
of
v
ariations
for
the
function
f
.
b
Complete
the
table
of
v
alues,
to
the
nearest
ten
th
:
x
−
4
−
3
−
2
−
1.5
−
1
−
0.8
f
(
x
)
x
−
0.3
0
0.5
1
2
3
4
f
(
x
)
3
Plot
the
curv
e
C
f
represen
ting
the
function
f
in
the
co-
ordinate
system
ab
o
v
e.
E.7465
Consider
the
function
f
defined
on
−
2
;
+
∞
b
y
the
relation
:
f
(
x
)
=
3
·
x
+8
x
+2
1
Establish
iden
tit
y
:
f
(
x
)
=
3+
2
x
+2
2
Establish
that
the
function
f
is
decreasing
on
−
2
;
+
∞
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/3025
sacados/3025
chapExoCorrec/440
sacados/440
chapExoCorrec/7333
sacados/7333
chapExoCorrec/4517
sacados/4517
-4
-3
-2
-1
2
3
4
I
-3
-2
-1
2
3
J
O
chapExoCorrec/7465
sacados/7465
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