1. Función Lineal - Escuela Colombiana de Ingeniería

ESCUELA COLOMBIANA DE INGENIERÍA
PRECÁLCULO 2003-2 MATERIAL ADICIONAL
FUNCION LINEAL
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) 8
7
6
5
4
3
2
1
-8 -7 -6 -5 -4 -3 -2 -1-1
-2
-3
-4
-5
-6
-7
-8
3.
<
65
4.
7 4 7 = > ?@ 7
A 5 = 5
7 C : 5 C ?D @ E 7
a.
<
b.
H
c.
J
d.
J
Q
b.
R
* + 1.
0
0
a.
b.
c.
= 7 A = 7 8 7 @ 4 5 E 5
, + - , ( x ,f ( x )) /
* + 1 2
1 f ( −2 )
a.
b.
c.
d.
2.
f (x)
, * + + . B : @ C ?D @
f (0 )
f ( −4 )
f (4)
* + 3
4 5 67 8 9 : 7
;
f (x) = 4
f ( x ) = −2
f ( x ) = −8
7 @ 7 6 5 @ 4 7 = ? F = A 65 @ F
C 5 = 4 7 8 ?5 @ F G 8 ? @ 7 @ C F @ 4 = 5 =
B : @ C ?D @ ;
F > ?@ ?F ;
5 @ I F ;
F F = E 7 @ 5 E 5 8 E 7 6 A : @ 4 F
E 7
C F = 4 7
C F @ 7 6 7 K 7
3 L
F F = E 7 @ 5 E 5 8 E 7 6 A : @ 4 F
E 7
C F = 4 7
C F @ 7 6 7 K 7
M L
?I 5 C : N 67 8 E 7
<
8 : = 7 8 A : 7 8 4 5 ;
a.
65
1 2 3 4 5 6 7 8
65
(
65 8 8 ?I : ? 7 @ 4 7 8 5 B ? = > 5 C ? F @ 7 8
6 A : @ 4 F
E 7
5
4 ?7 @ 7
= 7 C 4 5
c.
f ( −6 ) < 3
d.
5
2f (2 ) ≥ −
2
C F F = E 7 @ 5 E 5 8
A 7 @ E ?7 @ 4 7
8 F @ O 7 = E 5 E 7 = 5 8 P
C : N 67 8 8 F @ B 5 68 5 8 G
( −6 ; 7 ) ∈ f ( x )
e.
f (0 ) ≤ f ( 2)
3
f.
S
g.
S
h.
R
?
−4 ≤ x ≤ 4
7 @ 4 F @ C 7 8
K : 8 4 ?B ?C 5 @ E F
−4 ≤ f ( x ) ≤ 4
2
C:\Archivos Pagina\2003_2\Material\14_Funcion lineal.doc
?
5
−2 ≤ f ( x ) ≤ 1
7 @ 4 F @ C 7 8
B : @ C ?D @ 7 8 C = 7 C ?7 @ 4 7
−5 ≤ x ≤ 1
∀ x ∈ℜ
5.
8
7
6
5
4
3
2
1
(x ,y )
7.
8.
= 7
(x , y )
65
I = N B ?C 5
?4
E 7
65
= 7 C 4 5
8 ?4
7
6F 8 A : @ 4 F 8
4 5 67 8 9 : 7
f (x) = 2
f (x) = 7
f ( x ) + 3 = −2
2f ( x ) = 2
F = 7
S
9 : 7
S
x =2
65
= 7 C 4 5
7
L
8 ?4
G A 5 = 5
6F 8 A : @ 4 F 8
5 = 5
7 6 6F
3
6F 8
7 6 6F
: 8 7
7 @ 7 6 7 K 7
f (x) < 0
7
: 8 7
( x ; f ( x ))
g (x )
= 7 A = 7 8 7 @ 4 7
C F 6F = 7 8
O 5 6F = 7 8 E 7
6 @ 7 5 8 F
3
4 5 67 8 9 : 7
6 ?C : 5 8 L
1 2 3 4 5 6 7 8
9.
S
?4
7
7 @ 7 6 7 K 7
f (x) ≤ 0
C F @ A : @ 4 F
7 @
65
E 7
F = ?I 7 @ 7 @
C : 5 E = C : 65
E ?B 7 = 7 @ 4 7 8
G A 5 = 5
M
7 8 4 F
6F 8 O 5 6F = 7 8
: 8 7
E 7
M
4 5 67 8 9 : 7
C = : C 7 8
G
: 8 5 @ E F
A
h ( x ) = 3g ( x )
b.
j (x ) = g (x ) + 1
c.
k (x ) = g (x + 3)
: N 6 7 8
J
A : @ 4 F 8
: N 6 7 8
J
A : @ 4 F 8
65
= 7 A = 7 8 7 @ 4 5 C ?D @ I = N B ?C 5
(g ( x ) , x )
65
= 7 A = 7 8 7 @ 4 5 C ?D @ I = N B ?C 5
( x ; −g ( x ))
4 5 67 8
5 8 4 7 = ?8 C F 8
5 = 7 C 4 5 E ? : K 5 E 5 7 @ 65 8 ?I : ? 7 @ 4 7
R
C : 5 E = C : 65 C F = = 7 8 A F @ E 7 5 : @ 5 B : @ C ? D @
6?@ 7 5 6
e.
f (x) < 0
-8 -7 -6 -5 -4 -3 -2 -1-1
-2
-3
-4
-5
-6
-7
-8
d.
F
a.
b.
c.
d.
8
7
6
5
4
3
2
1
a.
S
6F 8
P ;
1
y = − f (x )
2
y = f ( x − 2)
d.
6.
7
y = f (x ) + 1
c.
1 2 3 4 5 6 7 8
8 ?4
y = f (x )
a.
b.
-8 -7 -6 -5 -4 -3 -2 -1-1
-2
-3
-4
-5
-6
-7
-8
10.
@ 7 6 A 65 @ F C 5 = 4 7 8 ?5 @ F
Q
A : @ 4 F 8
4 5 67 8 9 : 7
E 7
6F 8
E 7
6F 8
ESCUELA COLOMBIANA DE INGENIERÍA
PRECALCULO 2003 MATERIAL ADICIONAL
FUNCION LINEAL
11. S
7 5
65
f (x )
I = N B ?C 5
I = N B ?C 5
E 7
= 7 B 7 = 7 @ C ?5 L
H
7 65 C ? F @ 7
C 5 E 5
: @ 5
E 7
65 8 7 C : 5 C ? F @ 7 8
8 ?I : ? 7 @ 4 7 8 C F @
C F = = 7 8 A F @ E ?7 @ 4 7 ;
J = 5 B L
F
B 7 C 4 F
Q
7 F > 4 = ?C F
10
4
5
8
y = f (x − 4)
6
4
1
y = f (x ) + 5
6
f(x)
2
y = 3f ( x )
−8
−6
−4
−2
y = −f ( x )
2
−2
6
8
2
−4
3
4
−6
y = f ( x + 5)
−8
y = f (− x )
12. Q
− 10
@ 7 6 8 ?I : ? 7 @ 4 7
A 65 @ F
8
C 5 = 4 7 8 ?5 @ F
E ? : K 7
y
a.
R
67 @ 7
6
x
−2
2
4
6
8
R
I = 5 B ?9 : 7
d.
Q
e.
Q f.
65
b.
@ C : 7 @ 4 = 7
A = 7 8 7
: 65
A : 7 E 7
?C 5 E F 8 5 @ 4 7 = ?F = > 7 @ 4 7
= 7 C 4 5
65
65 ;
g (x )
f ( x ) + g (x )
1
−8
c.
f (x )
1
2
−6
4 5
0
−4
:
8 ?I : ?7 @ 4 7
−1
−2
F 8 A : @ 4 F 8
65
L
−2
−0, 7
2
−4
g (x ) = 3
3
4
−6
P
f ( x ) = 2x
9 : 7
6F 8 C F @ 4 ?7 @ 7
7 C : 5 C ?D @ L E 7
7 C : 5 C ?D @ E 7
C F @ C 6: ?= E 7
65
6F
65
8 F @ C F 6 ? @ 7 5 67 8 7 @ 7 6 A 65 @ F
: 8 4 ?B ?9 : 7
8 : = 7 8 A : 7 8 4 5 L
C 5 = 4 7 8 ?5 @ F
= 7 C 4 5
= 7 C 4 5
5 @ 4 7 = ?F = C:\Archivos Pagina\2003_2\Material\14_Funcion lineal.doc
5 @ 4 7 = ?F =
6F 8
( x ; f ( x ) + g ( x ))
7 @ 7 6 5 @ 4 7 = ? F = A 65 @ F
@ : 7 O 5
@ : 7 O 5
7 A = 7 8 7 @ 4 7
H
A : @ 4 F 8
7 @ 4 = > ?@ F 8 E 7
f (x )
P
g (x )
<
7
8 7 = 6F G
10
65
13. Q
@ 7 6 8 ?I : ? 7 @ 4 7
A 65 @ F
C 5 = 4 7 8 ?5 @ F
E ? : K 7
y
8
P
f ( x ) = 2x
a.
65 F = 7 : @ 5 4 5
Q
65 8 B : @ C ? F @ 7 8
65
O 5 6F = 7 8 A 5 = 5
P
g (x )
x = −3
O 5 6F =
> N
?> F
2
4
6
8
3
f (x )
−2
>
: @ 5
G C F @
@ ?> F
g (x )
7 @
C F @
f ( x ) + g (x )
−4
−6
b.
−8
c.
F 8 A : @ 4 F 8
R
I = 5 B ?9 : 7 65
d.
Q
e.
Q g.
@ C : 7 @ 4 = 7
A = 7 8 7
: :
65
65
9 : 7
8 F @ C F 6 ? @ 7 5 67 8 : 8 4 ?B ?9 : 7
6F 8 C F @ 4 ?7 @ 7 G 7 @ 7 6 5 @ 4 7 = ? F = A 65 @ F
7 C : 5 C ?D @ L E 7
7 C : 5 C ?D @ E 7
A : 7 E 7
7 @ 7 6 A 65 @ F
5 @ 4 7 = ?F =
6F 8
( x ; f ( x ) + g ( x ))
?C 5 E F 8 5 @ 4 7 = ?F = > 7 @ 4 7
= 7 C 4 5
7 A = 7 8 7 @ 4 7
H
A : @ 4 F 8
65
C F @ C 6: ?= E 7
7 A = 7 8 7 @ 4 7 I = N B ?C 5 > 7 @ 4 7
H
7 C : 5 C ?D @ E 7
65
@ : 7 O 5
@ : 7 O 5
6F
8 : = 7 8 A : 7 8 4 5 L
C 5 = 4 7 8 ?5 @ F
<
7
8 7 = 6F G
= 7 C 4 5
= 7 C 4 5
5 @ 4 7 = ?F = J
7 @ 8 : C : 5 E 7 = @ F
7 @ 4 = > ?@ F 8 E 7
F > A 5 = 7
C F @
65
f (x )
P
= 7 8 A : 7 8 4 5
g (x )
E 5 E 5
65 8 8 ?I : ?7 @ 4 7 8 B : @ C ? F @ 7 8 L
7 @
= 5 B ?9 : 7
B
P
7 @ C : 7 @ 4 = 7
h (x )
a.
b.
G
f ( x ) = −2 x + 1
f ( x ) = −2 x + 1
d.
f (x) =
G
1
x −1
2
G
G
G
8 ? 65
Q
8 : > 5
h ( x ) = f (x ) + g ( x )
g (x ) = 3
P
h ( x ) = f (x ) + g ( x )
1
x +1
2
x −2
g (x ) = −
2
g (x ) =
f ( x ) = 3x g (x ) = 3
7 = ?B ?9 : 7
P
g (x ) = 3
c.
e.
f ( x ) = 3x
E 7
G
P
P
h ( x ) = f (x ) + g ( x )
h ( x ) = f (x ) + g ( x )
i ( x ) = −2 x + 1
B : @ C ?F @ 7 8
P
h (x ) = f (x ) + g (x ) + i (x )
f (x ) + g ( x ) = g (x ) + f (x )
C : > A 67
65
A = F A ?7 E 5 E
C F @ > : 4 5 4 ?O 5
@ I 7 @ 7 = 5 6 A F = E 7 B ? @ ?C ?D @
5 = 5
E F 8 B : @ C ?F @ 7 8
C : 5 67 8 9 : ?7 = 5
f (x )
P
g (x )
8 7
E 7 B ?@ 7
f ( x ) + g ( x ) = (f + g )( x )
E 7
65 8
1
2
x
−2
P
x =3
?@ C = 7 > 7 @ 4 F 8 E 7
2
7 @
C 5 E 5
f ( x ) + g (x )
8 ?I : ?7 @ 4 7 8 C 5 = 5 C 4 7 = 8 4 ? C 5 8 G O 5 6F =
4
−4
E 7
G
f (x )
6
−6
L
g (x ) = x + 1
65
ESCUELA COLOMBIANA DE INGENIERÍA
PRECALCULO 2003 MATERIAL ADICIONAL
FUNCION LINEAL
7
14. S
8 5
g (x )
L
5 I 5
7
Q
65
9 : 7
7 8
65
I = N B ?C 5
A F =
6F
> 7 @ F 8 E F 8 A 5 = 7 K 5 8
h (x )
@ C : 7 @ 4 = 7
I = N B ?C 5
E ? : K 5 E 5
C F = = 7 8 A F @ E ?7 @ 4 7
7 @ C 5 E 5
P
9 : 7
E 7
= 7 A = 7 8 7 @ 4 5
B : @ C ?F @ 7 8 9 : 7
: @ F
E 7
4
A 65 @ F
8
C 5 = 4 7 8 ?5 @ F
E ? : K 7
y
a.
R
67 @ 7
−2
2
4
6
8
e.
Q
f.
8 C = ? 5
: 65
A : 7 E 7
65
65 ;
g (x )
f (x ) − g (x )
b.
−8
@ C : 7 @ 4 = 7
141
−6
Q
f (x )
4 5
0
−4
8 ?I : ?7 @ 4 7
−1
−2
:
65
L
−1,5
x
7 A = 7 8 7 @ 4 7
H
A : @ 4 F 8
7 @ 7 6 A 65 @ F
5 @ 4 7 = ?F =
6F 8
( x ; f ( x ) − g ( x ))
?C 5 E F 8 5 @ 4 7 = ?F = > 7 @ 4 7
= 7 C 4 5
g (x ) = 1
9 : 7
6F 8 C F @ 4 ?7 @ 7
7 C : 5 C ?D @ L E 7
65
8 F @ C F 6 ? @ 7 5 67 8 : 8 4 ?B ?9 : 7
7 @ 7 6 5 @ 4 7 = ? F = A 65 @ F
@ : 7 O 5
7 C : 5 C ?D @ E 7
65
@ : 7 O 5
C F @ C 6: ?= E 7
6F
5 @ 4 7 = ?F = = 7 C 4 5
= 7 C 4 5
C:\Archivos Pagina\2003_2\Material\14_Funcion lineal.doc
P
6 6N > 7 65
8 : = 7 8 A : 7 8 4 5 L
C 5 = 4 7 8 ?5 @ F
m (x )
7 @ 4 = > ?@ F 8 E 7
f (x )
P
g (x )
P
C F @ E ?C ?D @ E 5 E 5 P
−3
2
−4
P
3
4
d.
65
f (x )
4
f ( x ) = 3x
6
F 8 A : @ 4 F 8
R
I = 5 B ?9 : 7 65
B : @ C ?F @ 7 8
-4
@ 7 6 8 ?I : ?7 @ 4 7
c.
E F 8
C : > A 65 @ C F @
-4
-4
−6
E 7
4
-4
Q
8 : > 5
6F 8 A 65 @ F 8 C 5 = 4 7 8 ?5 @ F 8
4
15.
65
<
7
8 7 = 6F G
g.
F = 5
6 67 @ 7
3
65
8 ?I : ?7 @ 4 7
4 5
65 ;
g (x ) − f (x )
−3
−1,5
−1
0
141
h.
F > A 5 = 7 6F 8 = 7 8 : 64 5 E F 8 E 7
J
9 : 7 A : 7 E 7 C F @ C 6: ?= i.
H
16. Q
7 A = 7 8 7 @ 4 7
65 8
C F 6: > @ 5 8
6F 8 A : @ 4 F 8 7 @ 7 6 A 65 @ F P
@ 7 6 8 ?I : ? 7 @ 4 7
A 65 @ F
8
C 5 = 4 7 8 ?5 @ F
g (x ) − f (x )
I = 5 B ?9 : 7
E ? : K 7
y
f.
P
65 8 4 5
p (x ) = g (x ) − f (x )
g (x ) = −x + 2
65 F = 7 : @ 5 4 5
Q
65 8 B : @ C ? F @ 7 8
65
f (x )
LG 7 8
p (x ) = m ( x )
G O 5 6F =
> N
?> F
O 5 6F = 7 8 A 5 = 5
P
g (x )
7 @ 3
P
C 5 E 5
f (x) − g (x)
C F @
>
: @ 5
G C F @
@ ?> F
?@ C = 7 > 7 @ 4 F 8 E 7
1
2
x
−2
2
4
6
8
3
f (x )
−2
g (x )
f (x ) − g (x )
−4
−6
−8
7 A = 7 8 7 @ 4 7
g.
H
h.
F 8 A : @ 4 F 8
R
I = 5 B ?9 : 7 65
i.
Q
j.
Q @ C : 7 @ 4 = 7
A = 7 8 7
65
7 @ 7 6 A 65 @ F
:
6F 8 A : @ 4 F 8
?C 5 E F 8 5 @ 4 7 = ?F = > 7 @ 4 7
= 7 C 4 5
65
5 @ 4 7 = ?F =
9 : 7
( x ; f ( x ) − g ( x ))
8 F @ C F 6 ? @ 7 5 67 8 : 8 4 ?B ?9 : 7
6F 8 C F @ 4 ?7 @ 7 G 7 @ 7 6 5 @ 4 7 = ? F = A 65 @ F
7 C : 5 C ?D @ L E 7
7 C : 5 C ?D @ E 7
65
65
@ : 7 O 5
@ : 7 O 5
8 : = 7 8 A : 7 8 4 5 L
C 5 = 4 7 8 ?5 @ F
= 7 C 4 5
= 7 C 4 5
7 @ 4 = > ?@ F 8 E 7
f (x )
P
g (x )
E 7
65 8
7 @ 3
2
L
E 7
G
65 8 5 @ 4 7 = ? F = 7 8 G
8 ?I : ?7 @ 4 7 8 C 5 = 5 C 4 7 = 8 4 ? C 5 8 G O 5 6F =
4
−4
E 7
f (x ) − g (x )
7 C : 5 C ?D @
f ( x ) = −2 x
6
−6
65
P
<
7
8 7 = 6F G
ESCUELA COLOMBIANA DE INGENIERÍA
PRECALCULO 2003 MATERIAL ADICIONAL
FUNCION LINEAL
j.
: A : 7 E 7
C F @ C 6: ?= E 7
6F
7 A = 7 8 7 @ 4 7 I = N B ?C 5 > 7 @ 4 7
H
7 C : 5 C ?D @ E 7
7 @ 8 : C : 5 E 7 = @ F
h ( x ) = f (x ) − g ( x )
a.
G
1
x +2
2
f (x) =
b.
f (x ) = −x − 5
c.
f ( x ) = 2x + 1
d.
5 @ 4 7 = ?F = f ( x ) = −3 x + 1
Q
17.
S
7
8 5
g (x )
L
5 I 5
Q
65
7
18. R
5
C F > F
a.
<
65
65
7 @ C : 7 @ 4 = 7
65
m (x ) = f (x ) + g (x )
@ I 7 @ 7 = 5 6 A F = E 7 B ? @ ?C ?D @
5 = 5
E F 8 B : @ C ?F @ 7 8
9 : 7
7 8
m (x )
A F =
65
6F
I = N B ?C 5
f (x )
E ? : K 5 E 5
P
> 7 @ F 8 E F 8 A 5 = 7 K 5 8
7 @ C 5 E 5
P
g (x )
9 : 7
E 7
: @ F
8 7
E 7 B ?@ 7
= 7 A = 7 8 7 @ 4 5
65
B : @ C ?F @ 7 8 9 : 7
E 7
f ( x ) − g ( x ) = (f − g )( x )
= 7 8 4 5
2
2
4
−4
−2
2
−2
−2
−4
−4
= 7 A = 7 8 7 @ 4 5 E 5
f (x ) =
I = N B ?C 5
8 ?I : ?7 @ 4 7
4 5
67 5
7 @
3
x +3
5
65
C : 5 E = ? C : 65
65
f (x )
4
9
C F > A 67 4 5 =
6
65 ;
3
M
−9
−6
−3
3
−3
−6
−9
C:\Archivos Pagina\2003_2\Material\14_Funcion lineal.doc
P
C F @ E ?C ?D @ E 5 E 5 P
4 ?7 @ 7
L
6F 8 O 5 6F = 7 8 A 5 = 5
E F 8 B : @ C ?F @ 7 8
6F 8 A 65 @ F 8 C 5 = 4 7 8 ?5 @ F 8
4
2
E 7
C : > A 65 @ C F @
4
−2
3
C : 5 67 8 9 : ?7 = 5
C F = = 7 8 A F @ E ?7 @ 4 7
7 C : 5 C ?D @
7
P
g ( x ) = −3 x − 4
I = N B ?C 5
= 7 C 4 5
= 5 B ?9 : 7
g ( x ) = −2 x − 1
@ C : 7 @ 4 = 7
−4
2
x
3
g (x ) =
G
65 8 8 ?I : ?7 @ 4 7 8 B : @ C ? F @ 7 8 L
g ( x ) = −3
G
G
P
6
9
67 @ 7
R
6: 7 I F
b.
65
:
8 ?I : ?7 @ 4 7
?9 : 7
4 5
e.
Q f.
?8 4 7
J
19. J
A 65 @ F
?5 @ E F
65
5
C 5 = 4 7 8 ?5 @ F
E 7
6 67 I 5 = 5
65
I = N B ?C 5
7 C : 5 C ?D @ E 7
7 @ C : 7 @ 4 5
F > A 67 4 7
9 : ∀x ∈ ℜ
@ 4 ?A F
5 8 4 5
: h.
65
5 6I
7 @ ?7 @ E F
f (x )
g.
5 =
@ C : 7 @ 4 = 7
Q
?@ 4 7 = C 5 >
8 C ?8 5
P
65
F = E 7 @ 5 E 5
65 8 A 5 = 7 K 5 8 F
E 7
65
4 5
65
5 @ 4 7 = ?F = P
4 7 @ ?E 5 8 L
M
6 I 7 @ 7 = 5 6?
d.
65
7 @ 7 6 > ?8 > F
3
c.
8 7
F
4 ?7 @ 7 R
6N > 7 65
g (x )
8 ?> 7 4 = 5
7 @ 4 = 7
f (x )
P
g (x )
L
J
: N 6
8 : C F @ C 6: 8 ?D @ 5 @ 4 7 = ?F = 4 = 5 @ 8 B F = > 7
65
E 7
8 ?I : ? 7 @ 4 7
4 5
L
g (x )
g (x )
5 6I 7
g (f (0 )) =
g (0 ) =
f (g (0 )) =
f ( −5 ) =
g (f ( −5 )) =
g (3 ) =
f (g (3 )) =
f (10 ) =
g (f (10 )) =
g (6 ) =
f ( g (6 )) =
C F @ C 6: ?=
E 7
65
7 C : 5 C ?D @ E 7
65
f (0 ) =
A : 7 E 7
= 5 ?C 5 > 7 @ 4 7
L
6F 8 = 7 8 : 64 5 E F 8 F
4 7 @ ?E F 8 7 @
65
4 5
65
5 @ 4 7 = ?F = F > A 67 4 7
a.
S
b.
Q
?
65 8 8 ?I : ?7 @ 4 7 8 A = F A F 8 ? C ? F @ 7 8
a×b<0
G 7 @ 4 F @ C 7 8 G
6 C F @ K : @ 4 F
a<0
8 F 6: C ?D @ E 7
: @ 5
P
?@ 7 C : 5 C ?D @ 7 8
: @
D
?7 @
a>0
P
D
: @ 5
: @ ?D @ E 7
c.
? A 5 = 5 4 F E F
S
C F @ = 7 8 A 7 C 4 F
d.
Q
e.
Q
@
65 8 A 5 = 7 K 5 8
( x , y )∈ f ( x )
5 6
G 7 6 A : @ 4 F
( − x ,− y )∈ f ( x )
G 7 @ 4 F @ C 7 8 G 65
I = N B ?C 5
7 8 8 ? > 4 = ?C 5
F = E 7 @ 5 E 5 8 E 7 6 A 65 @ F
C 5 = 4 7 8 ?5 @ F
G P
5
65
3
8 7
67
6 65 > 5
@ B F = > 5
I 7 @ 7 = 5 6 C : 5 69 : ?7 = A : @ 4 F
8 F
= 7
65
I = N B ?C 5
E 7
: @ 5
B : @ C ?D @
f (x )
7 8
(
,
)
L
ESCUELA COLOMBIANA DE INGENIERÍA
PRECALCULO 2003 MATERIAL ADICIONAL
FUNCION LINEAL
f.
S
g.
Q
?
65
f (x ) = 2 x + 1
6
Q
6
S
?
5 6 F = ?I 7 @ 7 8
P
65
F = E 7 @ 5 E 5
E 7
5 6 F = ?I 7 @ 7 8
: @ 5
B : @ C ?D @ C F = = 7 8 A F @ E 7
5 6
E 7
: @ 5
B : @ C ?D @ C F = = 7 8 A F @ E 7
5 6
L
i.
8 C ?8 5
L
h.
5
f (− x ) = f ( x )
∀x ∈ ℜ
G 7 @ 4 F @ C 7 8
8 7
E ?C 7
9 : 7
65
B : @ C ?D @ 7 8
8 ?> 4 = ?C 5
= 7 8 A 7 C 4 F
5 6
j.
S
?
f (− x ) = − f ( x )
G 7 @ 4 F @ C 7 8 8 7
∀x ∈ ℜ
E ?C 7
9 : 7
65
B : @ C ?D @ 7 8 8 ?> 4 = ?C 5
= 7 8 A 7 C 4 F
5 6
k.
J
65
l.
R
m.
S
: 5 @ E F
7 @
: @ 5
B : @ C ?D @ 7 8
5
A 7 @ E ?7 @ 4 7
E 7
? C F @ F C 7 > F 8
65
5 = 7 C 4 5
R
C : 5 @ E F
9 : 7
: @ 5
= 7 C 4 5
I = N B ?C 5
5
n.
> 5 P F = O 5 6F = E 7 3 G 8 7
B : @ C ?D @ 5
F
4 ?7 @ 7
: @ O 5 6F =
> 5 P F = 7 @ M
G 8 7
E ?C 7
9 : 7
65
A 5 8 5
O 7 = 4 ?C 5 6 7 8
E 7
G 7 @ 4 F @ C 7 8
f( x )
(a , b )
P
4 ?7 @ 7
(c , d )
I = N B ?C 5
E 7
A F =
G 65
f( x )
C F > F
E 7
G
A 7 @ E ?7 @ 4 7
C F = = 7 8 A F @ E 7
f(x − 2 )
5
: @ 5
8 ?7 > A = 7
P
a≠c
o.
R
5
7 C : 5 C ?D @
E 7
B F = > 5
p.
Q
: @ 5
= 7 C 4 5
G > ?7 @ 4 = 5 8
9 : 7
65
6
C F 7 B ?C ?7 @ 4 7
B F = > 5
y = ax + b
s.
: @ 5
A : 7 E 7
= 7 C 4 5
7 8 C = ? ?= 8 7
A : 7 E 7
7 @
5 = 5
@ 5
65
B F = > 5
7 8 C = ? ?= 8 7
3
E 7
7 @
: @ 5
C F = = 7 8 A F @ E 7
5
7 C : 5 C ?D @ E 7
65
C F = = 7 8 A F @ E 7
5
7 @
65
B F = > 5
P
: @ 5
8 ? 7 8
E 7
> 7 = F
E 7
65
5 = 5 D @ 7 @ 4 = 7 7 6 @ > 7 = F E 7 : @ ? E 5 E 7 8 E 7 E 7 8 A 65 5 > ?7 @ 4 F O 7 = 4 ? C 5 6 P 7 6 @
R
: @ ? E 5 E 7 8 E 7 E 7 8 A 65 5 > ?7 @ 4 F
F = ? F @ 4 5 6 A 7 = > ?4 7 E 7 4 7 = > ? @ 5 =
B : @ C ?D @
r.
E 7
ax + by + c = 0 con a, b, c ∈ ℜ y a ≠ 0 y b ≠ 0
q.
8 ?7 > A = 7
7 C : 5 C ?D @
7 @
B : @ C ? D @ C : 5 69 : ?7 = 5 G
B : @ C ?D @ E 7 B ? @ ?E 5
A F =
f (x )
> N 8 E 7
7 8
65
: @ 5
7 C : 5 C ?D @ 8 7
6 65 > 5
P
3
7 8
65
B : @ C ?D @
t.
65
? C F @ F C 7 > F 8
S
> 7 E ?5 @ 4 7 : @ 5
65
I = N B ?C 5
E 7
f( x )
G
C:\Archivos Pagina\2003_2\Material\14_Funcion lineal.doc
7 @ 4 F @ C 7 8
E 7
f( x )
G
G
65
5 C ?5
I = N B ?C 5
E 7
f( x ) +1
8 7
F
4 ?7 @ 7
L
u.
5 = 5
E 7 4 7 = > ?@ 5 =
?I : 5 6 5
v.
?
f x <
S
7 8 4 N A F =
( )
7 @
g (x )
= 5 ?C 5 > 7 @ 4 7
C 7 = F G > ?7 @ 4 = 5 8
?I : 5 6 5
C 7 = F
: @
65
I = N B ?C 5
E 7
7 6 ?@ 4 7 = C 7 A 4 F
9 : 7
A 5 = 5
3
E 7
: @ 5
E 7 4 7 = > ?@ 5 =
B : @ C ?D @
f (x )
7 6 ?@ 4 7 = C 7 A 4 F
M
8 7
8 7
E 7
E 7
7
7
5 C 7 =
5 C 7 =
L
? @ 4 7 = O 5 6F
? C F @ F C 7 > F 8
S
: @ w.
5 6I 7
(a; b )
E 7
65
E 7
f( x )
G 7 @
I = N B ?C 5
f( x )
65
= 7 A = 7 8 7 @ 4 5 C ?D @
E 7
G 7 @ 4 F @ C 7 8
g (x )
G 65
I = N B ?C 5
7 @ 7 6 ? @ 4 7 = O 5 6F
I = N B ?C 5
E 7
8 7
(a; b )
3 f(x )
8 7
F
8 7 = O 5
F
4 ?7 @ 7
9 : 7
L
f (x )
> 7 E ?5 @ 4 7
L
20. En el siguiente sistema de coordenadas dibujar
f( x )=
2
3
x −1
Observar la gráfica y responder:
a. Para qué valores de x, es
y
8
f( x ) = 3
6
b. Escriba la expresión algebraica que repre senta
la anterior situación.
4
2
x
−6
−4
−2
2
4
6
8
−2
c.
Marque sobre la gráfica los puntos cuya
ordenada es menor o igual a 3 (Use asteriscos)
−4
d. Para qué valores de x, es f ( x ) ≤ 3 .(Márquelos
en la gráfica con líneas oblicuas)
−6
−8
e. Escriba la expresión algebraica que representa la anterior situación.
21. En el siguiente sistema de coordenadas, dibujar la gráfica de
8
y
f( x )=
2
3
x −1
y
g( x ) = 3
Observar las gráficas y contestar:
6
a. Para qué valor(es) de x son iguales las dos
funciones. Determine las coordenadas de ese (esos)
punto.
4
2
x
−6
−4
−2
2
−2
−4
−6
−8
4
6
8
b. Escribir la expresión algebraica que representa
la anterior situación
c. A partir de la gráfica determine: Para qué valores
de x es f ( x ) mayor que g ( x ) ?
ESCUELA COLOMBIANA DE INGENIERÍA
PRECALCULO 2003 MATERIAL ADICIONAL
FUNCION LINEAL
d. Escribir la expresión algebraica que represen ta la anterior situación:
e. Encuentre algebraicamente la solución.
f.
A partir de la gráfica determine Para qué valores de
x
es
menor que g( x ) ?
f(x)
g. Escribir la expresión algebraica que representa la anterior situación:
h. Encuentre algebraicamente la so lución
22. En el siguiente sistema de coordenadas, hacer la gráfica de
f ( x ) = −2 x + 3
y
g( x ) =
1
2
x −2
Observar las gráficas y contestar:
8
a. En qué puntos son iguales las dos funciones.
Dar la respuesta en la forma ( x , y )
y
6
4
b. Escribir la expresión algebraica que
representa la anterior situación
2
x
−6
−4
−2
2
4
6
8
−2
−4
c.
Para qué valores de
x
−6
es
f(x)
mayor que g ( x ) ?
−8
d. Escribir la expresión algebraica que
representa la anterior situación.
e. Encuentre algebraicamente la solución .
f.
Para qué valores de
x
es
f(x)
menor que g( x ) ?
g. Escribir la expresión algebraica que representa la anterior situación.
h. Encuentre algebraicamente la solución
23.
Q
@
: @ 5
O 7 @ 4 5
E 7
F E 7 I 5
4 F E F 8
C:\Archivos Pagina\2003_2\Material\14_Funcion lineal.doc
6F 8 5 = 4 C : 6F 8 4 ?7 @ 7 @
: @
E 7
E 7 8 C : 7 @ 4 F L
24.
25.
a.
8 C = ? 5 : @ 5 7 C : 5 C ?D @ A 5 = 5
Q
F = ?I ? @ 5 6 E 7 A 7 8 F 8 L
b.
J
c.
<
d.
: N 6 7 8 7 6 E F > ?@ ?F P
7
R
65
5
I = N B ?C 5
Q
b.
J
c.
J
d.
8 C = ? 5
: N 6 7 8
S
O 5 6F = 7 8
a.
<
b.
S
26. Q
65
: A F @I 5
b.
c.
5 C 7
4 7 = = 7 @ F
9 : 7
8 7
= 5
E 7
5
F 8
8 7
: 4 ?7 > A F
F 8 C F > A 5
5 8
65
H
J
d.
J
e.
J
B : @ C ?D @ L
7 6 C F 8 4 F
E 7
5 = = 7 @ E 5 > ?7 @ 4 F
E 7
: @ C 5 = = F A F =
E
C 5 @ 5
65
B : @ C ?D @ L
65
5
C F > F
@
> 7 = F
= 7 C 4 5 L
65 @
5 @
3
: @
7 8 7 6 @
B ?K F
E 7
> 7 = F
E 7
O 7 C 7 8 L
? 3 7 8 65 8 : > 5 E 7 6 F 8
S
O 7 C 7 8 9 : 7 65 8 : > 5 3 5 A 5 = 7 C 7 L
E 7
O 7 C 7 8 C : 5 67 8 8 F @
6F 8 7 8 4 : E ?5 @ 4 7 8
E 7
6F 8 A F 8 ? 67 8 O 5 6F = 7 8 A 5 = 5
3
L L L 7 @ 65 8 A = : 7 5 8 E ?5 I @ D 8 4 ? C 5 8 G
5
Q J G 7 6 A : @ 4 5 K 7
D @ C F @ 8 4 5 @ 4 7 7 @ 6F 8
64 ? > F 8 5 F 8 L
@
A = F > 7 E ?F E 7
Q
A : @ 4 F 8 G > ?7 @ 4 = 5 8 9 : 7 7 @ 7 6 5 F B : 7 E 7 A : @ 4 F 8 L
@ 4 = 5
D P
67
: @ 5
> N 8
O 7 @ 4 5 8
29.
65
E ?C
65
: @ 4 7 = = 7 @ F A F =
> 5 9 : ?@ 5
4 ?7 @ 7
C F > A 5
5
: 8 4 7 E
A 5 I 5
C F > ?8 ?D @ E 7 6
L
L
O 5 6F = E 7
: N 6 7 8 7 6 8 5 65 = ? F
: N 6 C F > A 5
: 5 6 F B 7 = 4 5
5 K 5 E F =
C F > ?8 ?D @ E 7
E 7
L
E 7
65
E D 65 = 7 8 P
F 8
: @ 4 F = @ F
6?@ 7 5 6> 7 @ 4 7 G
E D 65 = 7 8 P
C F > A = 5
7 @
65
65
A F =
5 8 4 5
> N 9 : ?@ 5
5 6 > F > 7 @ 4 F
Q
E 7
: @ 5
B N
A F =
: @ ?E 5 E
= ?C 5
C F > ?8 ?D @ P
: @ 5
L
6F 8 E F 8
E D 65 = 7 8 L
7 6 A : @ 4 F
E 7
7 @ 9 : 7
L
9 : 7
E D 65 = 7 8 L
?7 @ 7 8
65
R
F 8 4 = 5
C F > A 5
5 K F 8
5
8 F @ 7 8 7 @ C ?5 6 > 7 @ 4 7
A 5 I 5
C F > A 5
5
= 5 = 5
L
O 7
7 6
A F =
4 ?7 @ 7
8 7 > 5 @ 5
8 ?4 : 5 C ? D @ A 65 @ 4 7 5 E 5
65 8 E F 8 C F > A 5
> N 8 8 ? 8 7
P
7 @ O 7 @ 4 5 8 L
6 > 7 K F = O 7 @ E 7 E F = 7 @ C 5 E 5
C F = = 7 8 A F @ E ?7 @ 4 7
A 5 I 5
5 C 7 A 4 5 = 5
E 7
5
O 7 @ 4 5 8
: @ 7 > A 67 F
7 6
L
7 A = 7 8 7 @ 4 5 = I = N B ?C 5 > 7 @ 4 7
9 : E 7 A = 7 C ?D G 5 >
: @ O 5 6F =
E 7 6 > F > 7 @ 4 F
F B = 7 C 7 @ 5
> 5 P F = 7 8 E 7
5 = 5
8 7
L
5
O 5 6F = > ?8 > F G A 7 = F
c.
A 7 = 4 7 @ 7
E 7 6 E 5 E F P
4 = 5 @ 8 C : = = ?D
8 7 > 5 @ 5
b.
5
7 6 4 7 = = 7 @ F
a.
: @ O 5 6F =
B : @ C ?D @ L
E 7
F = ?8 C F > A = D
4 7 @ 5 @ 7 6 > ?8 > F
28. <
4 7 @ 5
8 ?4 : 5 C ? D @ A 65 @ 4 7 5 E 5 L
E F 8 E 5 E F 8 8 7
65 @
O 5 6F = ?
5 C 4 : 5 6> 7 @ 4 7
8 ? 7 8 4 7
A = 7 8 7 7 6 A : @ 4 5 K 7 A = F > 7 E ?F 7 @ B : @ C ?D @ E 7 6 4 ?7 > A F L
Q ? 65 4 7 @ E 7 @ C ?5 C F @ 4 ? @ 5 G C : N 6 8 7 = N 7 6 A : @ 4 5 K 7 A = F > 7 E ? F E 7 6F 8 7 8 4 : E ?5 @ 4 7 8 7 @ 7 6 S
F @ 65 > ? 8 > 5 4 7 @ E 7 @ C ? 5 G 7 @ 9 : 7 5 F 7 6 A : @ 4 5 K 7 A = F > 7 E ? F 8 7 = N E 7 A : @ 4 F 8 J
a.
27.
5
B : 7
65
: @ 5 = 4 C : 6F
A 7 @ E ?7 @ 4 7 L
A = F > 7 E ?F
E ?8 > ?@ : ?E F
B : @ C ?D @
E 7
7 6 E F > ?@ ?F
? 7 6 E 5 E F
5
65
> F 8 4 = 5 E F 8 5 = = ? 5
7 4 7 = > ?@ 7
E 7
65
65
O 7 @ 4 5
? 6D > 7 4 = F 8 = 7 C F = = ?E F 8 L
A 7 @ E ?7 @ 4 7
8 ?I @ ? B ? C 5
6 A : @ 4 5 K 7
A = : 7
E 7
65
E 7
= 7 A = 7 8 7 @ 4 5
E F 8 A 5 = 7 K 5 8 9 : 7
: N 6 7 8 7 6 = 5 @I F
: 7
? 6: 8 4 = 5
I = N B ?C 5
> 7 = F
E 7
C : > A 65 @ C F @
9 : 7
8 ?I : ?7 @ 4 7
B : @ C ?D @ E 7 6 @
a.
= 5 @I F
4 = 7 8 A 5 = 7 K 5 8 9 : 7
5 I 5
7 6 A = 7 C ?F
5 8 A 5 I 5 @
6F
> ?8 > F : 8 4 ?B ?9 : 7
8 : = 7 8 A : 7 8 4 5
8 : A 7 = 5 @
65 8 O 7 @ 4 5 8 C F = = 7 8 A F @ E ?7 @ 4 7 8 5
8 7
A : @ 4 F A F = 9 : ;
= 7 C ? 7
: @ 8 5 65 = ? F
A = F E : C ?E 5
5 6 E
5 L
E 7
L
E ?5 = ?F 8 L
E 7 > N 8 = 7 C ? 7
: @ 5
Q
6
ESCUELA COLOMBIANA DE INGENIERÍA
PRECALCULO 2003 MATERIAL ADICIONAL
FUNCION LINEAL
a.
8 C = ? 5 : @ 5 7 C : 5 C ? D @ 9 : 7 = 7 65 C ? F @ 7
Q
E 7 : @ ?E 5 E 7 8 A = F E : C ?E 5 8 A F = E 5 L
b.
<
c.
30.
: N @ 4 5 8
J
C 5
7 4 7 = > ?@ 7
R
H
67 8 G E F 8
A 7 @ E ?7 @ 4 7
E 7
Q
b.
Q
c.
Q
d.
Q
e.
S
: @ ?E 5 E 7 8 E 7
7
5 K 5 E F = 7 @
A = F E : C ?= 7 @
5
2
E 7
5
6F 8 C 5
67 8 7 @ 7 6 A ? 8 F
7 8 E 7
8 F
= 7
7 6 A ?8 F
A 5 = 5
7 6 C 5
@ C : 7 @ 4 = 7
65
5 64 : = 5
8 F
= 7
7 6 A ?8 F
A 5 = 5
7 6 C 5
@ C : 7 @ 4 = 7
65
A 7 @ E ?7 @ 4 7
@ C : 7 @ 4 = 7
65
6F @ I ?4 : E
? 8 7
E 7 C ?E 7
9 : 7
E 7
4 = 5
9 : 7
A = F E : C 7
9 : ?7 = 5
I 5 @ 5 =
5 K 5 E F = C F @ 7 6 @
> 7 = F
: @ ?E 5 E 7 8 L
L
5 64 : = 5
?> F
5
9 : 7
65
F B = 7 C 7
@ C : 7 @ 4 = 7
7 8 A F 8 5
A = D
: @ E
E 7 6 4 = 5
@ 5 5 @ 4 7 @ 5 E 7 4 7 67 O ? 8 ? D @ 7 8 4 N 8 ?7 @ E F 8 F A F = 4 5 E 5 A F = C : 5 4 = F I = : A F 8 E 7
J E 7 6F 8 C : 5 67 8 8 7 > : 7 8 4 = 5 @ 7 @ 7 6 I = N B ? C F L
6 > N 8 C F = 4 F E 7 65 ? 9 : ?7 = E 5 4 ?7 @ 7 : @ 5
Q
E 7 3 P 7 6 > N 8 65 = I F > ? E 7
A ?7 8 L
5 E ? 8 4 5 @ C ?5 E 7 65 5 8 7 E 7 65 5 @ 4 7 @ 5 5 6 A : @ 4 F
R
E 7 6 C 5
E 7 6 C 5
5 C F = 4 5 = 7 6 C 5
5 64 : = 5
E 7
65
67
4 F = = 7
7 E = F
5 K F P
67
5 4 5 9 : ?O 5
A 65 @ 4 7 5
8 7
67
67
65 = I F
E 7
67
67
65
> N 8 C F = 4 F L
> N 8
?
65 = I F L
9 : ?7 = E 5 L
> N 8 C F = 4 F L
> N 8
C F = 4 F
7 8 4 5 = N
? 8 7 E 7 C ? E 7 5 65 = I 5 = 7 6 C 5 67
S
B ?K 5 G 5 9 : 7 5 64 : = 5 E 7 65 4 F = = 7
f.
: @ E
5
B ?K 5 G 5
31. R
E 7 6 4 = 5
A F = E
5 > 5 = = 7
a.
7 6 8 5 65 = ? F
7 6 8 5 65 = ? F
> N 8
C 5 >
6 67 I 5 @ E F
65 = I F
C 5 >
?5 @ E F
7 6 C 5
?5 @ E F
65
A 7 @ E ?7 @ 4 7
5
67 65
A 7 @ E ?7 @ 4 7
5
5
4
G E 7 K 5 @ E F
G E 7 K 5 @ E F
65
65
5 8 7
5 8 7
6 67 I 5 = N E 7 E ?C 5
5
65
O 7 @ 4 5
E 7
8 F B 4 5 = 7 L
65 8 8 ?I : ? 7 @ 4 7 8 F A C ? F @ 7 8 A 5 = 5
5 7 > A = 7 8 5 67
R
8 : C F @ 4 = 5 4 5 C ?D @ 5 A 5 = 4 ?= E 7 6
> 7 8 L
a.
: 7 6E F
S
7 6 > 7 8 L
b.
J
5 8 7
> 7 @ 8 : 5 6 E 7
F > ?8 ?D @ E 7 6
8 F
= 7
65 8
L
> 5 8
O 7 @ 4 5 8
= 7 5 6?
E 7
C F > ?8 ?D @ 8 F
= 7
65 8 O 7 @ 4 5 8 = 7 5 6 ?
5 E 5 8 7 @
5 E 5 8 7 @ 7 6 > 7 8 L
? C 5 E 5 A 5 9 : 7 4 7 4 ? 7 @ 7 : @ O 5 6 F = E 7 L L 8 4 7 E 9 : 7 7 8 7 8 4 : E ?5 @ 4 7 E 7 = 7 C N 6 C : 6 F 67
S
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H
b.
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4 ?A F
E 7
E 7
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B : @ C ?D @ = 7 A = 7 8 7 @ 4 5
C 5 E 5
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