1. Lista 2 de C´lculo I
a 2010-2 3
UFES - Universidade Federal do Esp´
ırito Santo Opera¸˜es com fun¸˜es
co co
DMAT - Departamento de Matem´tica
a
Fun¸˜o composta
ca
LISTA 2 - 2010-2 Transforma¸˜es em gr´ficos
co a
1
1. Se f (x) = 3x2 + 2 e g(x) = , determine:
3x + 2
(a) (f + g)(x) (c) (f · g)(x) g
(e) (x)
f f
(b) (f (x))−1 (d) (x) (f) (f ◦ g)(x)
g
3−x
2. Seja f (x) = . Determine:
x
(a) f x2 − (f (x))2 1 1 (c) (f ◦ f )(x)
(b) f −
x f (x)
1
−x , x < 0 , x<0 (a) (f ◦ g)(x)
3. Dadas f (x) = e g(x) = √x , determine:
x2 , x ≥ 0 x , x≥0 (b) (g ◦ f )(x)
Nos exerc´
ıcios 4. a 11., a partir do gr´fico da fun¸˜o y = f (x) dado abaixo, esboce o gr´fico da
a ca a
fun¸˜o dada.
ca
y = f(x)
4. y = f (|x|) 8. y = f (x + 2) 4
2
9. y = f (x) + 3
5. y = |f (x)| –8 –6 –4 –2 0 2 4 6 8 10 12 14
x
f (x) + |f (x)| –2
10. y = –4
6. y = f (−x) 2
f (x) − |f (x)|
7. y = −f (x) 11. y =
2
Esboce os gr´ficos das fun¸˜es dos exerc´
a co ıcios 12. a 17.
12. f (x) = 8 − 3 x
2 −1 14. f (x) = |x − 1|3 16. f (x) = (|x| − 1)3
√
13. f (x) = 2|x| − 6 15. f (x) = 1 + 3 1 − x 17. f (x) = x2 − 4|x| + 3
Nos exerc´
ıcios 18. a 21. determime o dom´
ınio, a imagem e esboce o gr´fico da fun¸˜o dada.
a ca
18. f (x) = 3 sen (2πx) x 1
20. f (x) = sen − , 0 ≤ x ≤ 4π
2 2
x 1 π
19. f (x) = tan 21. f (x) = sec x −
2 2 3

2. Lista 2 de C´lculo I
a 2010-2 4
RESPOSTAS
9x3 + 6x2 + 6x + 5 3 2
1. (a) y = ,x = − (d) y = 9x3 + 6x2 + 6x + 4, x = −
3x + 2 2 3
1 1 2
(b) y = 2 (e) y = 3 ,x = −
3x + 2 9x + 6x2 + 6x + 4 3
3x2 + 2 2 18x2 + 24x + 11 2
(c) y = ,x = − (f) y = ,x = −
3x + 2 3 9x2 + 12x + 4 3
6x − 2x2 − 6 2
9x − 3x − 3 4x − 3
2. (a) y = (b) y = (c) y =
x2 3−x 3−x
√
1 −x , x < 0
− , x<0 (b) (g ◦ f )(x) =
3. (a) (f ◦ g)(x) = x x , x≥0
x , x≥0
4. 7. 10.
y y y
4 4 4
2 2 2
x x x
–14 –10 –8 –6 –4 –2 2 4 6 8 10 12 14 –14 –10 –8 –6 –4 –2 2 4 6 8 10 12 14 –14 –10 –8 –6 –4 –2 2 4 6 8 10 12 14
–2 –2 –2
–4 –4 –4
5. 8. 11.
y y
y
4 4
4
2 2
2
x x
–14 –10 –8 –6 –4 –2 2 4 6 8 10 12 14 –14 –10 –8 –6 –4 –2 2 4 6 8 10 12 14 x
–2 –2 –14 –10 –8 –6 –4 –2 2 4 6 8 10 12 14
–2
–4 –4
–4
6. 9.
y y
4 6
2 4
x 2
–14 –10 –8 –6 –4 –2 2 4 6 8 10 12 14
–2 x
–14 –10 –8 –6 –4 –2 0 2 4 6 8 10 12 14
–4 –2
12. 14. 16.
y y y
8 2 2
6
1
4 1
x
2
x –4 –3 –2 –1 1 2 3 4
10 20
x
–2 2 4 –1
–2
13. 15. 17.
y
8
y y
2
6 4
4 x
0 2
2 –4 –2 2 4 6
x x
–10 –8 –6 –4 –2 0 2 4 6 8 10 –2 –6 –4 –2 2 4 6
18. 20.
y
3
2
2
1
–6 –4 –2 0 2 4 x6
0 x
5 10
–2
19. 21.
y y
10 10
5
x x
–10 10 –15 –10 –5 5 10 15
–5
–10 –10