OpenStax Calculus Volume 1 – Instructor Solution Guide | Answers & Step-by-Step Calculus Study Resource 2026/2027

This OpenStax Calculus Volume 1 – Instructor Answer and Solution Guide provides structured solutions and explanations to support understanding of core calculus concepts. It covers key topics such as limits, derivatives, and integrals, helping clarify problem-solving methods used in introductory university calculus. This resource is ideal for students and instructors who want detailed, step-by-step solutions to strengthen comprehension and improve performance in calculus courses. Chapter 1 Functions and Graphs 1.1 Review of Functions Section Exercises For the following exercises, (a) determine the domain and the range of each relation, and (b) state whether the relation is a function. 1. x y x y –3 9 1 1 –2 4 2 4 –1 1 3 9 0 0 Answer: a. Domain =   − − − 3, 2, 1, 0,1 , 2, 3 , range =   0,1 , 4, 9 b. Yes, a function 2. x y x y –3 –2 1 1 –2 –8 2 8 –1 –1 3 –2 0 0 Answer: a. Domain =   − − − 3, 2, 1, 0,1 , 2, 3 , range =   − − − 2, 8, 1, 0,1 , 8 b. Yes, a function 3. x y x y 1 –3 1 1 2 –2 2 2 3 –1 3 3 0 0 Answer: a. Domain =   0,1 , 2, 3 , range =   − − − 3, 2, 1, 0,1 , 2, 3 b. No, not a function 4. x y x y 1 1 5 1 2 1 6 1 3 1 7 1 4 1 Answer: a. Domain =   1,2,3,4,5,6,7 , range =  1 b. Yes, a function OpenStax Calculus Volume 1 Instructor Answer and Solution Guide 5. x y x y 3 3 15 1 5 2 21 2 8 1 33 3 10 0 Answer: a. Domain =   3, 5, 8,1 0,1 5, 21, 33 , range =   0,1 , 2, 3 b. Yes, a function 6. x y x y –7 11 1 –2 –2 5 3 4 –2 1 6 11 0 –1 Answer: a. Domain =   −− 7, 2,0,1 , 3, 6 , range =   −− 1, 2,1 , 4, 5,1 1 b. No, not a function For the following exercises, find the values for each function, if they exist, then simplify. a. f (0) b. f (1) c. f (3) d. fx () − e. fa() f. f a h () + 7. f x x ( ) =− 52 Answer: a. −2 b. 3 c. 13 d. −− 52 x e. 52 a − f. 552 ah +− 8. ( ) 2 f x x x = − + 4 3 1 Answer: a. 1 b. 2 c. 28 d. 2 4 3 1 xx ++ e. 2 4 3 1 aa −+ f. 22 4 8 4 3 3 1 a ah h a h + + − − + 9. ( ) 2 fx x = Answer: a. Undefined b. 2 c 2 3 d. 2 x − e 2 a f. 2 ah + 10. f x x ( ) = − + 78 Answer: a. 15 b. 14 c. 12 d. − − + x 78 e. a −+ 78 f. ah + − + 78 11. f x x ( ) =+ 65 Answer: a. 5 b. 11 c. 23 d. −+ 65 x e. 65 a + f. 6 6 5 ah ++ 12. ( ) 2 37 x fx x − = + Answer: a. 2 7 − b. 1 10 − c. 1 16 d. 2 73 x x + − − e. 2 37 a a − + f. 2 337 ah ah +− ++ OpenStax Calculus Volume 1 Instructor Answer and Solution Guide 13. fx( ) = 9 Answer: a. 9 b. 9 c. 9 d. 9 e. 9 f. 9 For the following exercises, find the domain, range, and all zeros/intercepts, if any, of the functions. 14. ( ) 2 16 x fx x = − Answer: x 4; all real numbers; == 0; 0 y 15. g x x ( ) =− 81 Answer: 11 ; 0; ; 88 x y x   = no y-intercept 16. ( ) 2 3 4 hx x = + Answer: All real numbers; 3 0 4  y ; no x-intercept; 3 4 y = 17. f x x ( ) = − + + 12 Answer: x y x y  −  − = − = − + 2; 1; 1; 1 2 18. ( ) 1 9 fx x = − Answer: xy  9; 0; no intercepts 19. ( ) 3 4 gx x = − Answer: xy  4; 0; no x-intercept; 3 4 y =− 20. f x x ( ) =+ 45 Answer: All real numbers; y x y  = − = 0; 5; 20 21. ( ) 7 5 gx x = − Answer: xy  5; 0; no intercepts OpenStax Calculus Volume 1 Instructor Answer and Solution Guide For the following exercises, sketch the graph with the aid of the tables given: 22. ( ) 2 f x x =+1 x y x y –3 10 1 2 –2 5 2 5 –1 2 3 10 0 1 Answer: 23. f x x ( ) =− 36 x y x y –3 –15 1 –3 –2 –12 2 0 –1 –9 3 3 0 –6 Answer: 24. ( ) 1 1 2 f x x =+ OpenStax Calculus Volume 1 Instructor Answer and Solution Guide x y x y –3 1 2 − 1 3 2 –2 0 2 2 –1 1 2 3 5 2 0 1 Answer: 25. f x x ( ) = 2 x y x y –3 6 1 2 –2 4 2 4 –1 2 3 6 0 0 Answer: OpenStax Calculus Volume 1 Instructor Answer and Solution Guide 26. ( ) 2 f x x =− x y x y –3 –9 1 –1 –2 –4 2 –4 –1 –1 3 –9 0 0 Answer: 27. ( ) 3 f x x = x y x y –3 –27 1 1 –2 –8 2 8 –1 –1 3 27 0 0 Answer: OpenStax Calculus Volume 1 Instructor Answer and Solution Guide For the following exercises, use the vertical line test to determine whether each of the given graphs represents a function. Assume that a graph continues at both ends if it extends beyond the given grid. If the graph represents a function, then determine the following for each graph: a. Domain and range b. x -intercept, if any (estimate where necessary) c. y -Intercept, if any (estimate where necessary) d. The intervals for which the function is increasing e. The intervals for which the function is decreasing f. The intervals for which the function is constant g. Symmetry about any axis and/or the origin h. Whether the function is even, odd, or neither 28. Answer: Not a function 29. Answer: Function; a. Domain: all real numbers, range: y  0 b. x =1 c. y = 1 d. −   10 x and 1   x e. −   − x 1 and 01 x f. Not constant g. y-axis h. Even OpenStax Calculus Volume 1 Instructor Answer and Solution Guide 30. Answer: Function; a. Domain: all real numbers, range: −  y 3 b. x ~ 0.25, 3.75 c. y =−1 d. −  x 2 e. 2    x f. Not constant g. No symmetry h. Neither 31. Answer: Function; a. Domain: all real numbers, range: −   1.5 1.5 y b. x = 0 c. y = 0 d. all real numbers e. None f. Not constant g. Origin h. Odd 32. Answer: Not a function OpenStax Calculus Volume 1 Instructor Answer and Solution Guide 33. Answer: Function; a. Domain: −    x , range: −   22 y b. x = 0 c. y = 0 d. −   22 x e. Not decreasing f. −   − x 2 and 2    x g. Origin h. Odd 34. Answer: Function; a. Domain: all real numbers, range: y  0 b. 0 x  c. y = 0 d. x  0 e. Not decreasing f. −  x 0 g. No symmetry h. Neither 35. Answer: Function; a. Domain: −   4 4, x range: −   44 y b. x =1.2 c. y = 4 d. Not increasing e. 04 x f. −   40 x g. No Symmetry h. Neither OpenStax Calculus Volume 1 Instructor Answer and Solution Guide For the following exercises, for each pair of functions, find a. fg + b. fg − c. fg d. fg/ . Determine the domain of each of these new functions. 36. f x x ( ) =+ 34, g x x ( ) =−2 Answer: a 42 x + ; all real numbers b. 26 x + ; all real numbers c. 2 3 2 8 xx −− ; all real numbers d. 34 ; 2 2 x x x +  − 37. f x x ( ) =−8, ( ) 2 g x x = 5 Answer: a. 2 5 8; xx +− all real numbers b. 2 − + − 5 8; xx all real numbers c. 32 5 40 ; xx − all real numbers d. 2 8 5 x x − ; x  0 38. ( ) 2 f x x x = + + 3 4 1, g x x ( ) =+1 Answer: a. 2 3 5 2 xx ++ ; all real numbers b. 2 33 xx + ; all real numbers c. 32 3 7 5 1; x x x + + + all real numbers d. 3 1; 1 xx +  − 39. ( ) 2 f x x =−9 , ( ) 2 g x x x = − − 23 Answer: a. −+ 2 6; x all real numbers b. 2 − + + 2 2 12; xx all real numbers c. 4 3 2 − + + − − x x x x 2 12 18 27; all real numbers d. 3 1 x x + − + ; x −1, 3 40. f x x ( ) = , g x x ( ) =−2 Answer: a. xx +− 2; x  0 b. xx −+ 2; x  0 c. 3 xx − 2; x  0 d. 2 x x − ; xx  2, 0 41. ( ) 1 fx 6 x =+ , ( ) 1 gx x = Answer: a. 2 6; x + x  0 b. 6; x  0 c. 2 61 ; xx + x  0 d. 6 1; x + x  0 For the following exercises, for each pair of functions, find a.( )( ) f g x ฀ and b. ( )( ) g f x ฀ Simplify the results. Find the domain of each of the results. 42. f x x ( ) = 3 , g x x ( ) =+5 Answer: a. 3 15; x + all real numbers b. 3 5; x + all real numbers 43. f x x ( ) =+ 4 , g x x ( ) =− 41 Answer: a. 4 3; x + all real numbers b. 4 15; x + all real numbers