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# 3_3 - Chapter 3 Functions and Their Graphs 3.3 1 2 3...

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Chapter 3 308 Functions and Their Graphs 3.3 Properties of Functions 1. 2 x 5 2. slope = y x = 8 - 3 3 - - 2 ( 29 = 5 5 = 1 3. y = 2 x 2 + 3 Test for symmetry: x - axis: Replace y by - y : - y = 2 x 2 + 3 y = - 2 x 2 - 3, which is not equivalent to y = 2 x 2 + 3. y Replace x by - x y = 2 - x ( 29 2 + 3 = 2 x 2 + 3, which equivalent to y = 2 x 2 + 3. Origin: Replace x by - x and y by - y - y = 2 - x ( 29 2 + 3 y = - 2 x 2 - 3, which is not equivalent to y = 2 x 2 + 3. Therefore, the graph is symmetric with respect to the y -axis. 4. y - y 1 = m x - x 1 ( 29 y - - 2 ( 29 = 5 x - 3 ( 29 y + 2 = 5 x - 3 ( 29 5. y = x 2 - 9 y - intercept : Let x = 0, then y = 0 2 - 9 = - 9 (0, - 9) x - intercepts: Let y = 0, then 0 = x 2 - 9 x 2 = 9 x = ± 3 ( - 3,0), 3,0 ( 29 6. increasing 7. even, odd 8. True 9. True 10. False 11. Yes 12. No, it is increasing. 13. No, it only increases on (5, 10). 14. Yes 15. f is increasing on the intervals - 8, - 2 ( 29 , 0,2 ( 29 , 5, ( 29 . 16. f is decreasing on the intervals: -∞ , - 8 ( 29 , - 2,0 ( 29 , 2,5 ( 29 .

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Section 3.3 Properties of Functions 309 17. Yes. The local maximum at x = 2 is 10. 18. No. There is a local minimum at x = 5. 19. f has local maxima at x = - 2 and x = 2. The local maxima are 6 and 10, respectively. 20. f has local minima at x = - 8, x = 0 and x = 5. The local minima are –4, 0, and 0, respectively. 21. (a) Intercepts: (–2,0), (2,0), and (0,3). (b) Domain: x - 4 x 4 { } ; Range: y 0 y 3 { } . (c) Increasing: (–2, 0) and (2, 4); Decreasing: (–4, –2) and (0, 2). (d) Since the graph is symmetric with respect to the y -axis, the function is even. 22. (a) Intercepts: (–1, 0), (1, 0), and (0, 2). (b) Domain: x - 3 x 3 { } ; Range: y 0 y 3 { } . (c) Increasing: (–1, 0) and (1, 3); Decreasing: (–3, –1) and (0, 1). (d) Since the graph is symmetric with respect to the y -axis, the function is even. 23. (a) Intercepts: (0,1). (b) Domain: x x is any real number { } ; Range: y y > 0 { } . (c) Increasing: ( -∞ , ) ; Decreasing: never. (d) Since the graph is not symmetric with respect to the y -axis or the origin, the function is neither even nor odd. 24. (a) Intercepts: (1, 0). (b) Domain: x x > 0 { } ; Range: y y is any real number { } . (c) Interval notation: Increasing: (0, ) ; Decreasing: never. (d) Since the graph is not symmetric with respect to the y -axis or the origin, the function is neither even nor odd. 25. (a) Intercepts: ( ,0),( π ,0), and (0,0) . (b) Domain: x -π≤ x ≤π { } ; Range: y - 1 y 1 { } . (c) Increasing: - π 2 , π 2 ; Decreasing: , - π 2 and π 2 , π . (d) Since the graph is symmetric with respect to the origin, the function is odd. 26. (a) Intercepts: - π 2 , 0 , π 2 , 0 , and (0, 1). (b) Domain: x x ; Range: y - 1 y 1 { } . (c) Increasing: ,0 ( 29 ; Decreasing: 0, π ( 29 . (d) Since the graph is symmetric with respect to the y -axis, the function is even.
Chapter 3 Functions and Their Graphs 310 27. (a) Intercepts: 1 2 ,0 , 5 2 , and 0, 1 2 .

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3_3 - Chapter 3 Functions and Their Graphs 3.3 1 2 3...

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