Chapter 1 A Review of Functions

Section 1.2: Functions and Graphs  Graphing a Function  Additional Properties of Functions and Their Graphs Graphing a Function Definition: The Vertical Line Test: 24 SECTION 1.2 Functions and Graphs Example: Solution: MATH 1330 Precalculus 25 CHAPTER 1 A Review of Functions Example: Solution: 26 SECTION 1.2 Functions and Graphs Example: Solution: MATH 1330 Precalculus 27 CHAPTER 1 A Review of Functions Additional Example 1: Solution: Part (a): Part (b): 28 SECTION 1.2 Functions and Graphs Additional Example 2: Solution: MATH 1330 Precalculus 29 CHAPTER 1 A Review of Functions 30 SECTION 1.2 Functions and Graphs Additional Example 3: Solution: MATH 1330 Precalculus 31 CHAPTER 1 A Review of Functions Additional Example 4: Solution: 32 SECTION 1.2 Functions and Graphs Additional Example 5: MATH 1330 Precalculus 33 CHAPTER 1 A Review of Functions Solution: 34 SECTION 1.2 Functions and Graphs Additional Example 6: Solution: MATH 1330 Precalculus 35 CHAPTER 1 A Review of Functions 36 SECTION 1.2 Functions and Graphs Additional Properties of Functions and Their Graphs An x-intercept of the Graph of a Function: The y-intercept of the Graph of a Function: MATH 1330 Precalculus 37 CHAPTER 1 A Review of Functions Finding the Intercepts of the Graph of a Function: Example: Solution: 38 SECTION 1.2 Functions and Graphs Increasing/Decreasing Functions: Maximum/Minimum Values of a Function: MATH 1330 Precalculus 39 CHAPTER 1 A Review of Functions Even/Odd Functions: 40 SECTION 1.2 Functions and Graphs MATH 1330 Precalculus 41 CHAPTER 1 A Review of Functions Example: Solution: 42 SECTION 1.2 Functions and Graphs Additional Example 1: Solution: MATH 1330 Precalculus 43 CHAPTER 1 A Review of Functions Additional Example 2: 44 SECTION 1.2 Functions and Graphs Solution: MATH 1330 Precalculus 45 CHAPTER 1 A Review of Functions Additional Example 3: Solution: 46 SECTION 1.2 Functions and Graphs Additional Example 4: Solution: MATH 1330 Precalculus 47 CHAPTER 1 A Review of Functions Additional Example 5: Solution: Additional Example 6: Solution: Additional Example 7: 48 SECTION 1.2 Functions and Graphs Solution: MATH 1330 Precalculus 49 Exercise Set 1.2: Functions and Graphs Determine whether or not each of the following graphs represents a function. 9. y x 1. y x y 10. 2. y x x 3. y x 4. y x 5. y (1, 5), 12. (3, 2), (1, 2), (0,  3), (2, 1), ( 2, 1) 13. Analyze the coordinates in each of the sets above. Describe a method of determining whether or not the set of points represents a function without graphing the points. 14. Determine whether or not each set of points represents a function without graphing the points. Justify each answer. (a) (7, 3), (3,  7), (1, 5), (5, 1), (2, 1) (b) y (c) (d) x 8. (2, 4), (3, 4), (2, 1), (3, 6) y x 7. 11. Answer the following. x 6. For each set of points, (a) Graph the set of points. (b) Determine whether or not the set of points represents a function. Justify your answer. (6, 3), (4, 3), (2, 3), (3, 3), (5, 3) (3, 6), (3,  4), (3, 2), (3,  3), (3, 5) (2,  5), (5, 2), (2, 5), (5,  2), (5, 2) y Continued on the next page... x 50 Exercise Set 1.2: Functions and Graphs (c) Find the y-intercept(s) of the function. Answer the following. 15. The graph of y  f (x) is shown below.  y f (d) Find the following function values: g (2); g (0); g (1); g (3); g (6) (e) For what value(s) of x is g ( x )  2 ?  (f) On what interval(s) is g increasing?  (g) On what interval(s) is g decreasing?  (h) What is the maximum value of the function?  (i) What is the minimum value of the function? x        17. The graph of y  g (x) is shown below.   y  (a) Find the domain of the function. Write your answer in interval notation.  (b) Find the range of the function. Write your answer in interval notation. g  (c) Find the y-intercept(s) of the function. x (d) Find the following function values:     f (2); f (0); f (4); f (6) (e) For what value(s) of x is f ( x)  9 ? (f) On what interval(s) is f increasing? (g) On what interval(s) is f decreasing? (h) What is the maximum value of the function? (i) What is the minimum value of the function?   (a) Find the domain of the function. Write your answer in interval notation. (b) Find the range of the function. Write your answer in interval notation. (c) How many x-intercept(s) does the function have? 16. The graph of y  g (x) is shown below. y (e) Which is greater, g (2) or g (3) ? g  (d) Find the following function values: g (2); g (0); g (2); g (4); g (6) (f) On what interval(s) is g increasing?  (g) On what interval(s) is g decreasing?  x      Continued on the next page… (a) Find the domain of the function. Write your answer in interval notation. (b) Find the range of the function. Write your answer in interval notation. MATH 1330 Precalculus 51 Exercise Set 1.2: Functions and Graphs 18. The graph of y  f (x) is shown below. 25. f ( x)  x  3 y 26. f ( x)  5  x  f  27. F ( x)  x 2  4 x  28. G ( x)  ( x  3)2  1 x      (a) Find the domain of the function. Write your answer in interval notation. (b) Find the range of the function. Write your answer in interval notation. 29. f ( x)  x3  1 30. f ( x)  x 4  16 31. g ( x)  12 x 32. h( x)   8 x (c) Find the x-intercept(s) of the function. (d) Find the following function values: f (3); f (2); f (1); f (1); f (4) (e) Which is smaller, f (0) or f (3) ? (f) On what interval(s) is f increasing? (g) On what interval(s) is f decreasing? For each of the following functions: (c) State the domain of the function. Write your answer in interval notation. (d) Find the intercepts of the function. (e) Choose x-values corresponding to the domain of the function, calculate the corresponding yvalues, plot the points, and draw the graph of the function. 19. f ( x)   32 x6 20. f ( x)  23 x  4 21. h( x)  3x  5, 1  x  3 22. h( x)  2x,  3  x  2 23. g ( x)  x  3 24. g ( x)  x  4 52 For each of the following piecewise-defined functions: (a) State the domain of the function. Write your answer in interval notation. (b) Find the y-intercept of the function. (c) Choose x-values corresponding to the domain of the function, calculate the corresponding yvalues, plot the points, and draw the graph of the function. 2 x  4, if  2  x  1  x  3, if 1  x  5 33. f ( x)    1 x  2, if  3  x  0 34. f ( x)   3  4 x  3, if x  0 3, if x  2  5, if x  2 35. f ( x)   4, if  5  x  1 2, if 1  x  3 36. f ( x)   4, if x  0  x 2  1, if x  0 37. f ( x)   3  x 2 , if x  1  3, if x  1 38. f ( x)   Exercise Set 1.2: Functions and Graphs  x, if x  3  39. f ( x)   x 2 , if  3  x  2 4, if x  2  40.  x 2  5, if x  0  f ( x)  1, if 0  x  3 2  x, if x  3  Answer the following. 41. (a) If a function is odd, then it is symmetric with respect to the __________. (x-axis, y-axis, or origin?) (b) If a function is even, then it is symmetric with respect to the __________. (x-axis, y-axis, or origin?) 42. (a) If a function is symmetric with respect to the y-axis, then the function is __________. (Odd, even, both, or neither?) (b) If a function is symmetric with respect to the origin, then the function is __________. (Odd, even, both or neither?) 43. Suppose that y  f (x) is an odd function and that (3, 6) is a point on the graph of f. Find another point on the graph. 44. Suppose that y  f (x) is an even function and that (2,  7) is a point on the graph of f. Find another point on the graph. Determine whether each of the following functions is even, odd, both or neither. 45. f ( x)  x3  5 x 46. f ( x)  x 2  3x 47. f ( x)  x 4  2 x 2 48. f ( x)  x5  2 x3 49. f ( x)  2 x3  x 2  5 x  1 50. f ( x)  3x6  2 x2 MATH 1330 Precalculus 53