Exercise List – Calculus 1:
Functions and Domains
1. Determine the domain and range of the function 𝑓(𝑥) = √(𝑥 − 2).
2. Let 𝑓(𝑥) = 1
(𝑥² − 4) be. What is the domain?
3. Check if 𝑓(𝑥) = |𝑥| is even, odd or neither.
4. Tell me if 𝑓(𝑥) = √(𝑥²) is equal to |𝑥|.
5. Give the domain of the function 𝑓(𝑥) = 𝑙𝑛(𝑥 − 1).
6. Sketch the graph of 𝑓(𝑥) = |𝑥 − 2|.
7. Find the intersection points of the function 𝑓(𝑥) = 𝑥² − 4 with the axles.
8. Check if 𝑓(𝑥) = 𝑥³ is even, odd or neither.
9. Sketch the graph of the piecewise function: 𝑓(𝑥) = 𝑥 + 1, if 𝑥 < 0; 𝑓(𝑥) =
𝑥², 𝑖𝑓 𝑥 ≥ 0.
10. For which value of a does the function𝑓(𝑥) = √(𝑥 − 𝑎) has domain [1, ∞)?
Limits e Continuity
11. Calculate lim
𝑥→2
(𝑥² − 4)
(𝑥 − 2) .
12. Calculate lim
𝑥→0
𝑠𝑖𝑛(𝑥)
𝑥 .
13. Check if there is the limit lim
𝑥→0−
1
𝑥 .
14. Calculate lim
𝑥→∞
(3𝑥² + 2)
(𝑥² − 1) .
15. Study the continuity of𝑓(𝑥) = 1
(𝑥 − 3).
16. Tell me if 𝑓(𝑥) = |𝑥| is continuous in 𝑥 = 0.
17. Use the Intermediate Value Theorem to show that 𝑓(𝑥) = 𝑥³ − 𝑥 − 1 has
roots in [1, 2].
18. Determine the discontinuity points of𝑓(𝑥) = (𝑥² − 1)
(𝑥 − 1) .
19. 19. Sketch a continuous function that is increasing at (−∞, 0), constant in (0,1)
and decreasing in (1, ∞).
20. Prove that the function 𝑓(𝑥) = 1
𝑥 is not continuous in 𝑥 = 0.
