Lista 2 de Calculo I 2010-2 3

UFES - Universidade Federal do Espırito Santo

DMAT - Departamento de Matematica

LISTA 2 - 2010-2

Operacoes com funcoes

Funcao composta

Transformacoes em graficos

1. Se f(x) = 3x2 + 2 e g(x) =1

3x + 2, determine:

(a) (f + g)(x)

(b) (f(x))−1

(c) (f · g)(x)

(d)

(

f

g

)

(x)

(e)

(

g

f

)

(x)

(f) (f ◦ g)(x)

2. Seja f(x) =3 − x

x. Determine:

(a) f(

x2)

− (f(x))2 (b) f

(

1

x

)

− 1

f(x)(c) (f ◦ f)(x)

3. Dadas f(x) =

{

−x , x < 0x2 , x ≥ 0

e g(x) =

{ 1

x, x < 0

√x , x ≥ 0

, determine:(a) (f ◦ g)(x)(b) (g ◦ f)(x)

Nos exercıcios 4. a 11., a partir do grafico da funcao y = f(x) dado abaixo, esboce o grafico dafuncao dada.

4. y = f (|x|)

5. y = |f(x)|

6. y = f(−x)

7. y = −f(x)

8. y = f(x + 2)

9. y = f(x) + 3

10. y =f(x) + |f(x)|

2

11. y =f(x) − |f(x)|

2

x

y = f(x)

–4

–2

0

2

4

–8 –6 –4 –2 2 4 6 8 10 12 14

Esboce os graficos das funcoes dos exercıcios 12. a 17.

12. f(x) = 8 − 3√

x

2− 1

13. f(x) =√

2|x| − 6

14. f(x) = |x − 1|3

15. f(x) = 1 + 3√

1 − x

16. f(x) = (|x| − 1)3

17. f(x) =∣

∣x2 − 4|x| + 3∣

∣

Nos exercıcios 18. a 21. determime o domınio, a imagem e esboce o grafico da funcao dada.

18. f(x) = 3 sen (2πx)

19. f(x) = tan(x

2

)

20. f(x) =

∣

∣

∣

∣

sen(x

2

)

− 1

2

∣

∣

∣

∣

, 0 ≤ x ≤ 4π

21. f(x) =1

2sec

(

x − π

3

)

Lista 2 de Calculo I 2010-2 4

RESPOSTAS

1. (a) y =9x3 + 6x2 + 6x + 5

3x + 2, x 6= −3

2

(b) y =1

3x2 + 2

(c) y =3x2 + 2

3x + 2, x 6= −2

3

(d) y = 9x3 + 6x2 + 6x + 4, x 6= −2

3

(e) y =1

9x3 + 6x2 + 6x + 4, x 6= −2

3

(f) y =18x2 + 24x + 11

9x2 + 12x + 4, x 6= −2

3

2. (a) y =6x − 2x2 − 6

x2(b) y =

9x − 3x2 − 3

3 − x(c) y =

4x − 3

3 − x

3. (a) (f ◦ g)(x) =

{

− 1

x, x < 0

x , x ≥ 0

(b) (g ◦ f)(x) =

{ √−x , x < 0x , x ≥ 0

4.y

x

–4

–2

2

4

–14 –10 –8 –6 –4 –2 2 4 6 8 10 12 14

5.y

x

–4

–2

2

4

–14 –10 –8 –6 –4 –2 2 4 6 8 10 12 14

6.y

x

–4

–2

2

4

–14 –10 –8 –6 –4 –2 2 4 6 8 10 12 14

7.y

x

–4

–2

2

4

–14 –10 –8 –6 –4 –2 2 4 6 8 10 12 14

8.y

x

–4

–2

2

4

–14 –10 –8 –6 –4 –2 2 4 6 8 10 12 14

9.y

x

–20

2

4

6

–14 –10 –8 –6 –4 –2 2 4 6 8 10 12 14

10.y

x

–4

–2

2

4

–14 –10 –8 –6 –4 –2 2 4 6 8 10 12 14

11.y

x

–4

–2

2

4

–14 –10 –8 –6 –4 –2 2 4 6 8 10 12 14

12.y

x–2

2

4

6

8

10 20

13.y

x0

2

4

6

8

–10 –8 –6 –4 –2 2 4 6 8 10

14.

x

y

1

2

–2 2 4

15.y

x

–2

0

2

–4 –2 2 4 6

16.

x

y

–1

1

2

–4 –3 –2 –1 1 2 3 4

17.y

x

2

4

–6 –4 –2 2 4 6

18.y

x

–2

0

2

–6 –4 –2 2 4 6

19.y

x

–10

10

–10 10

20.

x0

123

5 10

21.y

x

–10

–5

5

10

–15 –10 –5 5 10 15