lista_mat1_funcao_modular

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Matema´tica I (ENP0002)
Professor: Rebeca Dourado Gonc¸alves
Aluno:
Lista de Exerc´ıcio - Func¸a˜o Modular
1. Calcule:
a) |8− 12|
b) | − 3− 2|
c) | − 1|+ | − 3|
d) | − 4 + 6|
e) | − 3| − | − 2 + 1|
f) || − 4| − | − 2||
g) | − 3(−4)− 2(+8)|
h) | − 1
2
+ 1
3
− 1
2
+ 0, 5|
2. Resolva as equac¸o˜es modulares em R:
a) |2x + 3| = 9
b) |3x− 7| = 1
2
c) 2|x− 1| = 3
2
d) |x− 5|+ 1 = 2
e) | − 4x + 1| = 2
f) 2 + |3x− 6| = 8
g) |3− 4x| = −3
h) |x
2
+ 1| = 1
4
i)
√
x2 = 2
3
j)
√
x2 + 2x + 1 = 3
3. Dadas as equac¸o˜es a seguir, resolva-as
em R:
a) |2x| = |x− 3|
b) |1− x| = 1− x
c) |x + 1| = |2x− 3|
d) |x2 − 5x| = 14
e) |x|2 − 10|x|+ 21 = 0
f) |x2 + 2x− 15| = 0
g) |3x2 − x− 1| = 1
h) 3|x|2 − |x| − 2 = 0
4. Calcule o domı´nio das func¸o˜es reais a
seguir:
a) f(x) = 3x|x|−1
b) g(x) = 1|x|
c) h(x) = −2|x−2|
d) h(x) = 3x−1|2x−3|− 1
2
5. Resolva as inequac¸o˜es a seguir em R:
a) |x− 1| > 3
b) |2x− 4| ≥ 1
c) |x− 7| ≥ 0
d) 2|x− 1| > 1
e) |x2 − 3x| ≤ 1
f) |x− 3| > −2
g)
√
(x + 2)2 − 8x < 5
6. Dadas as inequac¸o˜es abaixo, resolva-as
e, R:
a) |x2 − x− 1| ≤ 1
b) (x2 − x)|x + 1| > 0
c) |x− 2|(x + 1) ≤ 0
d) 3x−1|2x−3| ≤ 0
e) |x|
2x− 1
2
≤ 0
f) x
2+3x
|3x−1| < 0
7. Calcule o domı´nio de cada func¸a˜o real a
seguir:
a) f(x) =
√|x− 1| − 1
b) f(x) = 1√|x|−1
8. Construa os gra´ficos e determine o
domı´nio e a imagem das func¸o˜es reais:
1
Matema´tica I (ENP0002)
a) f(x) = |x− 2|
b) y = | − 2x + 3|
c) f(x) = |x− 2| − 2
d) y = |x− 2|+ 2
e) y = −|x− 1|
f) y =
√
(x + 1)2
g) y = || − x + 2| − 1|
h) y = x|x|
i) f(x) = | − x2 + 2x + 3|
Gabarito
1.
a) 4
b) 5
c)4
d)2
e)2
f)2
g)4
h) 7
6
2.
a)S= {−6, 3}
b)S= { 5
2
, 13
6
}
c)S= { 1
4
, 7
4
}
d)S= {4, 6}
e)S= {− 1
4
, 3
4
}
f)S= {0, 4}
g) S= ∅
h)S= {− 3
2
,− 5
2
}
i)S= {− 2
3
, 2
3
}
j)S= {−4, 2}
3.
a)S= {−3, 1}
b)S= {x ∈ R|x ≤ 1}
c)S= { 2
3
, 4}
d)S= {−2, 7}
e)S= {−7,−3, 3, 7}
f)S= {−5, 3}
g) S= {− 2
3
, 0, 1
3
, 1}
h)S= {−1, 1}
4.
a)D(f)= R− {−1, 1}
b)D(g)= R∗
c)D(h) = R− {2}
d)D(i)= R− { 5
4
, 7
4
}
5.
a)S= {x ∈ R|x < −2 ou x > 4}
b)S= {x ∈ R|x ≥ 5
2
ou x ≤ 3
2
}
c)S= R
d)S= {x ∈ R|x > 3
2
ou x < 1
2
}
e) S= {x ∈ R| 3−
√
13
2
≤ x ≤ 3−
√
5
2
ou 3+
√
5
2
≤ x ≤ 3+
√
13
2
}
f)S= R
g) S= {x ∈ R| − 3 < x < 7}
6.
a) S= {x ∈ R| − 1 ≤ x ≤ 0 ou 1 ≤ x ≤ 2}
b) S= {x ∈ R|x < 0 ou x > 1 e x 6= −1}
c) S= {x ∈ R|x ≤ −1 ou x = 2}
d) S= {x ∈ R|x ≤ 1
3
}
e)S= {x ∈ R|x < 1
4
}
f)S= {x ∈ R| − 3 < x < 0}
7.
a) D(f)= (−∞, 0] ∪ [2,+∞)
b) D(f)= (−∞,−1[∪]1,+∞)
8.
a) D(f)= R; Im(f)= R+
b) D(f)= R; Im(f)= R+
c) D(f)= R; Im(f)= [−2,+∞)
d) D(f)= R; Im(f)= [2,+∞)
e) D(f)= R; Im(f)= R−
f) D(f)= R; Im(f)= R+
g) D(f)= R; Im(f)= R+
h) D(f)= R∗; Im(f)= [−1, 1]
i) D(f)= R; Im(f)= R+
2