Appunti Metodi Matematici e Statistici – Scheda Esercitazione N. 1 – Calcolo del Dominio

Ancona, 4 ottobre 2011 1

CALCOLARE IL DOMINIO DELLE SEGUENTI FUNZIONI f:RR

1. 𝑓 𝑥 = 𝑙𝑛2+𝑥

2−𝑥 𝑅. 𝐷𝑓 = 𝑥 ∈ 𝑹: − 2 < 𝑥 < 2 = −2, 2

2. 𝑓 𝑥 = −𝑥 + 1

3−𝑥 𝑅. 𝐷𝑓 = 𝑥 ∈ 𝑹: 𝑥 ≤ 0 = −∞, 0

3. 𝑓 𝑥 = 2𝑥2+ 3

𝑥− 𝑥2− 4 𝑅. 𝐷𝑓 = 𝑥 ∈ 𝑹: 𝑥 ≤ −2 ∨ 𝑥 ≥ 2 = −∞, −2 ∪ 2, +∞

4. 𝑓 𝑥 = 𝑥 − 2 + 2 2 − 𝑥 + 𝑥2 + 3 𝑅. 𝐷𝑓 = 𝑥 ∈ 𝑹: 𝑥 = 2 = 2

5. 𝑓 𝑥 = ln (1+𝑥)

𝑥−1 𝑅. 𝐷𝑓 = 𝑥 ∈ 𝑹: − 1 < 𝑥 < 1 ∨ 𝑥 > 1 = −1, 1 ∪ 1, +∞

6. 𝑓 𝑥 = arcsin(𝑙𝑛𝑥) 𝑅. 𝐷𝑓 = 𝑥 ∈ 𝑹: 1

𝑒≤ 𝑥 ≤ 𝑒 =

1

𝑒, 𝑒

7. 𝑓 𝑥 = 𝑎𝑟𝑐𝑐𝑜𝑠x−3

2− 𝑙𝑜𝑔10 4 − 𝑥 − 𝑥 − 1 𝑅. 𝐷𝑓 = 𝑥 ∈ 𝑹: 1 ≤ 𝑥 < 4 = 1, 4

8. 𝑓 𝑥 = 𝑥3𝑥

𝑥−3 𝑅. 𝐷𝑓 = 𝑥 ∈ 𝑹: 0 < 𝑥 < 1 ∨ 1 < 𝑥 < 3 ∨ 𝑥 > 3 = 0, 1 ∪ 1, 3 ∪ 3, +∞

9. 𝑓 𝑥 = x2 + x − 6 𝑅. 𝐷𝑓 = 𝑥 ∈ 𝑹: 𝑥 ≤ −2 ∨ 𝑥 ≥ 2 = −∞,−2 ∪ 2, +∞

10. 𝑓 𝑥 = ln 𝑥 − 2 𝑅. 𝐷𝑓 = 𝑥 ∈ 𝑹: 𝑥 < −2 ∨ 𝑥 > 2 = −∞,−2 ∪ 2, +∞

11. 𝑓 𝑥 = ln(𝑠𝑖𝑛𝑥) 𝑅. 𝐷𝑓 = 𝑥 ∈ 𝑹: 2𝑘𝜋 < 𝑥 < 𝜋 + 2𝑘𝜋, 𝑘 ∈ 𝒁

12. 𝑓 𝑥 = 1

ln 1−𝑥 + 𝑥 + 5 𝑅. 𝐷𝑓 = 𝑥 ∈ 𝑹: − 3 ≤ 𝑥 < 0 ∨ 0 < 𝑥 < 1 = −3, 0 ∪ 0, 1

13. 𝑓 𝑥 = 𝑙𝑛 𝑥 − 4 + 6 − 𝑥 𝑅. 𝐷𝑓 = 𝑥 ∈ 𝑹: 4 ≤ 𝑥 ≤ 6 = 4, 6

14. 𝑓 𝑥 = ln 𝑥2− 6𝑥+8

𝑥2− 2

3 𝑅. 𝐷𝑓 = 𝑥 ∈ 𝑹: − 3 ≤ 𝑥 < 0 ∨ 0 < 𝑥 < 1 =

= −∞,− 2 ∪ 2, 2 ∪ 4, +∞

15. 𝑓 𝑥 = ln 1 − ln(𝑥2 − 5𝑥 + 16) 𝑅. 𝐷𝑓 = ∅

16. 𝑓 𝑥 = 1 − 𝑙𝑜𝑔2|1 − 𝑥| 𝑅. 𝐷𝑓 = 𝑥 ∈ 𝑹: − 1 ≤ 𝑥 < 1 ∨ 1 < 𝑥 < 3 = −1, 1 ∪ 1,3

17. 𝑓 𝑥 = ln(2 − 𝑥 ) 𝑅. 𝐷𝑓 = 𝑥 ∈ 𝑹: − 1 ≤ 𝑥 ≤ 1 = −1, 1

Appunti Metodi Matematici e Statistici – Scheda Esercitazione N. 1 – Calcolo del Dominio

Ancona, 4 ottobre 2011 2

18. 𝑓 𝑥 = 𝑎𝑟𝑐𝑠𝑖𝑛 𝑙𝑛𝑥

10 𝑅. 𝐷𝑓 = 𝑥 ∈ 𝑹:

10

𝑒≤ 𝑥 ≤ 10𝑒 =

10

𝑒, 10𝑒

19. 𝑓 𝑥 = 1 − 2𝑥 + 3𝑎𝑟𝑐𝑠𝑖𝑛 3𝑥−1

2 𝑅. 𝐷𝑓 = 𝑥 ∈ 𝑹: −

1

3≤ 𝑥 ≤

1

2 = −

1

3,

1

2

20. 𝑓 𝑥 = arcsin log2x 𝑅. 𝐷𝑓 = 𝑥 ∈ 𝑹: 1 ≤ 𝑥 ≤ 2 = 1, 2

21. 𝑓 𝑥 = 𝑥−2|𝑥−1|

𝑥 𝑙𝑛 𝑥2 𝑅. 𝐷𝑓 = 𝑥 ∈ 𝑹: 0 < 𝑥 < 1 ∨ 𝑥 > 1 = 0, 1 ∪ 1, +∞