Practice Sheet (02) : Functions & Graphs Part A : Domain & Range (10 questions) 1. Find the domain of 𝑓 ( 𝑥 ) = 5 𝑥 − 1 𝑥 ! − 16 2. Find the domain: 𝑔 ( 𝑥 ) = √ 3 𝑥 − 9 3. Find the domain of the logarithmic function: ℎ ( 𝑥 ) = ln ( 2 − 𝑥 ) 4. Determine the domain and range of the constant function 𝑘 ( 𝑥 ) = 7 5. Determine the domain of 𝑚 ( 𝑥 ) = √ 𝑥 + 2 𝑥 − 3 6. State all real numbers for which the expression is defined: 7 25 − 𝑥 ! 7. Domain of the rational - absolute function: 𝑓 ( 𝑥 ) = ∣ 𝑥 + 1 ∣ 𝑥 ! − 1 8. Domain of 𝑎 ( 𝑥 ) = 7 2 𝑥 ! − 5 𝑥 − 3 9. Range of 𝑟 ( 𝑥 ) = 3 𝑥 ! + 2 10. Range of the piecewise function: 𝑝 ( 𝑥 ) = < 2 𝑥 + 1 𝑥 < 0 𝑥 ! 𝑥 ≥ 0 Part B : Function Values & Algebra (8 questions) 11. Evaluate 𝑓 ( 3 ) , 𝑓 ( √ 5 ) , and 𝑓 ( 𝑥 + ℎ ) for 𝑓 ( 𝑥 ) = 𝑥 ! − 4 𝑥 + 1 12. Simplify: 𝑔 ( 𝑥 + ℎ ) − 𝑔 ( 𝑥 ) , 𝑔 ( 𝑥 ) = 1 𝑥 13. Find 𝑘 ( 𝑎 ! ) and 𝑘 ( 2 𝑎 + ℎ ) for 𝑘 ( 𝑥 ) = √ 𝑥 + 5 14. Compute: 𝑚 ( 𝑥 + ℎ ) − 𝑚 ( 𝑥 ) , 𝑚 ( 𝑥 ) = 3 𝑥 − 7 15. Evaluate: 𝑞 ( 2 𝑡 + 1 ) , 𝑞 ( 𝑧 ! − 3 𝑧 ) , 𝑞 ( 𝑥 + ℎ ) , 𝑞 ( 𝑥 ) = 4 𝑥 ! 16. If 𝑓 ( 𝑥 ) = 𝑥 " − 3 𝑥 , find 𝑓 ( − 2 ) , 𝑓 ( 0 ) , and 𝑓 ( 2 + ℎ ) 17. If 𝑓 ( 𝑥 ) = ∣ 3 𝑥 − 1 ∣ find 𝑓 ( − 1 ) , 𝑓 E # " F , and 𝑓 ( 𝑥 + ℎ ) 18. Evaluate ℎ ( 𝑥 + ℎ ) − ℎ ( 𝑥 ) , ℎ ( 𝑥 ) = √ 𝑥 Part C : Difference Quotient (6 questions) 19. Compute 𝑓 ( 𝑥 + ℎ ) − 𝑓 ( 𝑥 ) ℎ , 𝑓 ( 𝑥 ) = 𝑥 ! + 2 𝑥 20. Compute DQ for 𝑓 ( 𝑥 ) = 5 𝑥 − 8 21. Compute 𝑔 ( 𝑥 + ℎ ) − 𝑔 ( 𝑥 ) ℎ , 𝑔 ( 𝑥 ) = 1 𝑥 − 4 22. Compute the difference quotient of 𝑓 ( 𝑥 ) = √ 𝑥 + 3 23. Compute DQ for 𝑚 ( 𝑥 ) = 𝑥 " 24. Compute 𝑘 ( 𝑥 + ℎ ) − 𝑘 ( 𝑥 ) ℎ , 𝑘 ( 𝑥 ) = 𝑥 𝑥 + 1 Part D : Intercepts & Graph Skills (6 questions) 25. Find x - and y - intercepts of 𝑦 = − 3 𝑥 + 9 26. Find all intercepts of 𝑦 = 2 𝑥 − 6 𝑥 + 4 27. Determine intercepts of 𝑥 = 5 28. Sketch the graph of 𝑦 = ∣ 𝑥 − 3 ∣ and label the vertex. 29. Find intercepts and sketch 𝑦 = 2 √ 𝑥 30. Graph the piecewise function and find intercepts: 𝑓 ( 𝑥 ) = < − 2 𝑥 + 1 𝑥 < 1 𝑥 ! − 4 𝑥 ≥ 1 Part E : One - to - One & Composition (8 questions) 31. Determine whether 𝑓 ( 𝑥 ) = 𝑥 " + 1 is one - to - one. 32. Is 𝑓 ( 𝑥 ) = 𝑥 ! − 2 one - to - one? Why or why not? 33. Find the composition: 𝑓 ( 𝑥 ) = 3 𝑥 − 2 , 𝑔 ( 𝑥 ) = 𝑥 ! Compute: • 𝑓 ( 𝑔 ( 𝑥 ) ) • 𝑔 ( 𝑓 ( 𝑥 ) ) 34. If 𝑓 ( 𝑥 ) = √ 𝑥 , 𝑔 ( 𝑥 ) = 2 𝑥 + 5 find 𝑓 ( 𝑔 ( 𝑥 ) ) and 𝑔 ( 𝑓 ( 𝑥 ) ) 35. Find ( 𝑓 + 𝑔 ) ( 𝑥 ) , ( 𝑓 − 𝑔 ) ( 𝑥 ) , ( 𝑓𝑔 ) ( 𝑥 ) , and $ % ( 𝑥 ) for: 𝑓 ( 𝑥 ) = 𝑥 − 1 , 𝑔 ( 𝑥 ) = 𝑥 + 3 36. Determine whether ℎ ( 𝑥 ) = 𝑥 𝑥 ! + 1 is one - to - one. 37. Find the composition: 𝑓 ( 𝑥 ) = 𝑥 ! + 4 , 𝑔 ( 𝑥 ) = 1 𝑥 Compute 𝑓 ( 𝑔 ( 𝑥 ) ) and 𝑔 ( 𝑓 ( 𝑥 ) ) 38. If 𝑓 ( 𝑥 ) = ∣ 𝑥 ∣ , 𝑔 ( 𝑥 ) = 𝑥 − 3 find 𝑓 ( 𝑔 ( 𝑥 ) ) Part F : Piecewise Functions (6 questions) 39. Evaluate the function at the given points: 𝑓 ( 𝑥 ) = < 2 𝑥 − 1 𝑥 < 0 𝑥 ! + 3 𝑥 ≥ 0 Compute: • 𝑓 ( − 2 ) • 𝑓 ( 0 ) • 𝑓 ( 3 ) 40. Graph the piecewise function: 𝑔 ( 𝑥 ) = I − 𝑥 𝑥 ≤ − 1 2 𝑥 + 1 − 1 < 𝑥 < 2 4 𝑥 ≥ 2 41. Determine whether the function is continuous at 𝑥 = 1 : 𝑓 ( 𝑥 ) = I 𝑥 + 2 𝑥 < 1 3 𝑥 = 1 2 𝑥 − 1 𝑥 > 1 42. For ℎ ( 𝑥 ) = L 𝑥 ! 𝑥 < 2 𝑥 + 5 𝑥 ≥ 2 Find: • ℎ ( 1 ) • ℎ ( 2 ) • ℎ ( 5 ) 43. Sketch 𝑚 ( 𝑥 ) = I 0 𝑥 < 0 𝑥 0 ≤ 𝑥 ≤ 4 4 𝑥 > 4 44. Graph and find intercepts of: 𝑝 ( 𝑥 ) = L 𝑥 " 𝑥 < 1 2 𝑥 − 3 𝑥 ≥ 1

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Practice Sheet 2 (1)