{"id":10855,"date":"2015-07-13T22:27:00","date_gmt":"2015-07-13T22:27:00","guid":{"rendered":"https:\/\/courses.candelalearning.com\/osprecalc\/?post_type=chapter&p=10855"},"modified":"2021-10-25T02:58:31","modified_gmt":"2021-10-25T02:58:31","slug":"domain-and-range-solutions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/chapter\/domain-and-range-solutions\/","title":{"raw":"Solutions: Domain and Range","rendered":"Solutions: Domain and Range"},"content":{"raw":"

Solutions for Odd-Numbered Section Exercises<\/span><\/h2>\r\n1.\u00a0The domain of a function depends upon what values of the independent variable make the function undefined or imaginary.\r\n\r\n3. \u00a0There is no restriction on [latex]x[\/latex] for [latex]f\\left(x\\right)=\\sqrt[3]{x}[\/latex] because you can take the cube root of any real number. So the domain is all real numbers, [latex]\\left(-\\infty ,\\infty \\right)[\/latex]. When dealing with the set of real numbers, you cannot take the square root of negative numbers. So [latex]x[\/latex] -values are restricted for [latex]f\\left(x\\right)=\\sqrt[]{x}[\/latex] to nonnegative numbers and the domain is [latex]\\left[0,\\infty \\right)[\/latex].\r\n\r\n5.\u00a0Graph each formula of the piecewise function over its corresponding domain. Use the same scale for the [latex]x[\/latex] -axis and [latex]y[\/latex] -axis for each graph. Indicate inclusive endpoints with a solid circle and exclusive endpoints with an open circle. Use an arrow to indicate [latex]-\\infty [\/latex] or [latex]\\text{ }\\infty [\/latex]. Combine the graphs to find the graph of the piecewise function.\r\n\r\n7.\u00a0[latex]\\left(-\\infty ,\\infty \\right)[\/latex]\r\n\r\n9.\u00a0[latex]\\left(-\\infty ,3\\right][\/latex]\r\n\r\n11.\u00a0[latex]\\left(-\\infty ,\\infty \\right)[\/latex]\r\n\r\n13.\u00a0[latex]\\left(-\\infty ,\\infty \\right)[\/latex]\r\n\r\n15.\u00a0[latex]\\left(-\\infty ,-\\frac{1}{2}\\right)\\cup \\left(-\\frac{1}{2},\\infty \\right)[\/latex]\r\n\r\n17.\u00a0[latex]\\left(-\\infty ,-11\\right)\\cup \\left(-11,2\\right)\\cup \\left(2,\\infty \\right)[\/latex]\r\n\r\n19.\u00a0[latex]\\left(-\\infty ,-3\\right)\\cup \\left(-3,5\\right)\\cup \\left(5,\\infty \\right)[\/latex]\r\n\r\n21.\u00a0[latex]\\left(-\\infty ,5\\right)[\/latex]\r\n\r\n23.\u00a0[latex]\\left[6,\\infty \\right)[\/latex]\r\n\r\n25.\u00a0[latex]\\left(-\\infty ,-9\\right)\\cup \\left(-9,9\\right)\\cup \\left(9,\\infty \\right)[\/latex]\r\n\r\n27. Domain: [latex]\\left(2,8\\right][\/latex] \u00a0\u00a0Range [latex]\\left[6,8\\right)[\/latex]\r\n\r\n29. Domain: [latex]\\left[-4, 4\\right][\/latex] \u00a0 Range: [latex]\\left[0, 2\\right][\/latex]\r\n\r\n31. Domain: [latex]\\left[-5,\\text{ }3\\right)[\/latex]\u00a0 \u00a0Range: [latex]\\left[0,2\\right][\/latex]\r\n\r\n33. Domain: [latex]\\left(-\\infty ,1\\right][\/latex] \u00a0 Range: [latex]\\left[0,\\infty \\right)[\/latex]\r\n\r\n35. Domain: [latex]\\left[-6,-\\frac{1}{6}\\right]\\cup \\left[\\frac{1}{6},6\\right][\/latex] \u00a0 Range: [latex]\\left[-6,-\\frac{1}{6}\\right]\\cup \\left[\\frac{1}{6},6\\right][\/latex]\r\n\r\n37. Domain: [latex]\\left[-3,\\text{ }\\infty \\right)[\/latex] \u00a0 Range: [latex]\\left[0,\\infty \\right)[\/latex]\r\n\r\n39. Domain: [latex]\\left(-\\infty ,\\infty \\right)[\/latex]\r\n\r\n\"Graph\r\n\r\n41. Domain: [latex]\\left(-\\infty ,\\infty \\right)[\/latex]\r\n\r\n\"Graph\r\n\r\n43. Domain: [latex]\\left(-\\infty ,\\infty \\right)[\/latex]\r\n\r\n\"Graph\r\n\r\n45. Domain: [latex]\\left(-\\infty ,\\infty \\right)[\/latex]\r\n\r\n\"Graph\r\n\r\n47.\u00a0[latex]\\begin{cases}f\\left(-3\\right)=1;& f\\left(-2\\right)=0;& f\\left(-1\\right)=0;& f\\left(0\\right)=0\\end{cases}[\/latex]\r\n\r\n49.\u00a0[latex]\\begin{cases}f\\left(-1\\right)=-4;& f\\left(0\\right)=6;& f\\left(2\\right)=20;& f\\left(4\\right)=34\\end{cases}[\/latex]\r\n\r\n51.\u00a0[latex]\\begin{cases}f\\left(-1\\right)=-5;& f\\left(0\\right)=3;& f\\left(2\\right)=3;& f\\left(4\\right)=16\\end{cases}[\/latex]\r\n\r\n53. Domain: [latex]\\left(-\\infty ,1\\right)\\cup \\left(1,\\infty \\right)[\/latex]\r\n\r\n55. Window: [latex]\\left[-0.5,-0.1\\right][\/latex] \u00a0 Range: [latex]\\left[4,\\text{ }100\\right][\/latex]\r\n\r\n\"Graph\r\n\r\nWindow: [latex]\\left[0.1,\\text{ }0.5\\right][\/latex] \u00a0 Range: [latex]\\left[4,\\text{ }100\\right][\/latex]\r\n\r\n\"Graph\r\n\r\n57.\u00a0[latex]\\left[0,\\text{ }8\\right][\/latex]\r\n\r\n59.\u00a0Many answers. One function is [latex]f\\left(x\\right)=\\frac{1}{\\sqrt{x - 2}}[\/latex].\r\n\r\n61.\u00a0The domain is [latex]\\left[0,\\text{ }6\\right][\/latex]; it takes 6 seconds for the projectile to leave the ground and return to the ground.\r\n

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Solutions for Odd-Numbered Section Exercises<\/span><\/h2>\n

1.\u00a0The domain of a function depends upon what values of the independent variable make the function undefined or imaginary.<\/p>\n

3. \u00a0There is no restriction on [latex]x[\/latex] for [latex]f\\left(x\\right)=\\sqrt[3]{x}[\/latex] because you can take the cube root of any real number. So the domain is all real numbers, [latex]\\left(-\\infty ,\\infty \\right)[\/latex]. When dealing with the set of real numbers, you cannot take the square root of negative numbers. So [latex]x[\/latex] -values are restricted for [latex]f\\left(x\\right)=\\sqrt[]{x}[\/latex] to nonnegative numbers and the domain is [latex]\\left[0,\\infty \\right)[\/latex].<\/p>\n

5.\u00a0Graph each formula of the piecewise function over its corresponding domain. Use the same scale for the [latex]x[\/latex] -axis and [latex]y[\/latex] -axis for each graph. Indicate inclusive endpoints with a solid circle and exclusive endpoints with an open circle. Use an arrow to indicate [latex]-\\infty[\/latex] or [latex]\\text{ }\\infty[\/latex]. Combine the graphs to find the graph of the piecewise function.<\/p>\n

7.\u00a0[latex]\\left(-\\infty ,\\infty \\right)[\/latex]<\/p>\n

9.\u00a0[latex]\\left(-\\infty ,3\\right][\/latex]<\/p>\n

11.\u00a0[latex]\\left(-\\infty ,\\infty \\right)[\/latex]<\/p>\n

13.\u00a0[latex]\\left(-\\infty ,\\infty \\right)[\/latex]<\/p>\n

15.\u00a0[latex]\\left(-\\infty ,-\\frac{1}{2}\\right)\\cup \\left(-\\frac{1}{2},\\infty \\right)[\/latex]<\/p>\n

17.\u00a0[latex]\\left(-\\infty ,-11\\right)\\cup \\left(-11,2\\right)\\cup \\left(2,\\infty \\right)[\/latex]<\/p>\n

19.\u00a0[latex]\\left(-\\infty ,-3\\right)\\cup \\left(-3,5\\right)\\cup \\left(5,\\infty \\right)[\/latex]<\/p>\n

21.\u00a0[latex]\\left(-\\infty ,5\\right)[\/latex]<\/p>\n

23.\u00a0[latex]\\left[6,\\infty \\right)[\/latex]<\/p>\n

25.\u00a0[latex]\\left(-\\infty ,-9\\right)\\cup \\left(-9,9\\right)\\cup \\left(9,\\infty \\right)[\/latex]<\/p>\n

27. Domain: [latex]\\left(2,8\\right][\/latex] \u00a0\u00a0Range [latex]\\left[6,8\\right)[\/latex]<\/p>\n

29. Domain: [latex]\\left[-4, 4\\right][\/latex] \u00a0 Range: [latex]\\left[0, 2\\right][\/latex]<\/p>\n

31. Domain: [latex]\\left[-5,\\text{ }3\\right)[\/latex]\u00a0 \u00a0Range: [latex]\\left[0,2\\right][\/latex]<\/p>\n

33. Domain: [latex]\\left(-\\infty ,1\\right][\/latex] \u00a0 Range: [latex]\\left[0,\\infty \\right)[\/latex]<\/p>\n

35. Domain: [latex]\\left[-6,-\\frac{1}{6}\\right]\\cup \\left[\\frac{1}{6},6\\right][\/latex] \u00a0 Range: [latex]\\left[-6,-\\frac{1}{6}\\right]\\cup \\left[\\frac{1}{6},6\\right][\/latex]<\/p>\n

37. Domain: [latex]\\left[-3,\\text{ }\\infty \\right)[\/latex] \u00a0 Range: [latex]\\left[0,\\infty \\right)[\/latex]<\/p>\n

39. Domain: [latex]\\left(-\\infty ,\\infty \\right)[\/latex]<\/p>\n

\"Graph<\/p>\n

41. Domain: [latex]\\left(-\\infty ,\\infty \\right)[\/latex]<\/p>\n

\"Graph<\/p>\n

43. Domain: [latex]\\left(-\\infty ,\\infty \\right)[\/latex]<\/p>\n

\"Graph<\/p>\n

45. Domain: [latex]\\left(-\\infty ,\\infty \\right)[\/latex]<\/p>\n

\"Graph<\/p>\n

47.\u00a0[latex]\\begin{cases}f\\left(-3\\right)=1;& f\\left(-2\\right)=0;& f\\left(-1\\right)=0;& f\\left(0\\right)=0\\end{cases}[\/latex]<\/p>\n

49.\u00a0[latex]\\begin{cases}f\\left(-1\\right)=-4;& f\\left(0\\right)=6;& f\\left(2\\right)=20;& f\\left(4\\right)=34\\end{cases}[\/latex]<\/p>\n

51.\u00a0[latex]\\begin{cases}f\\left(-1\\right)=-5;& f\\left(0\\right)=3;& f\\left(2\\right)=3;& f\\left(4\\right)=16\\end{cases}[\/latex]<\/p>\n

53. Domain: [latex]\\left(-\\infty ,1\\right)\\cup \\left(1,\\infty \\right)[\/latex]<\/p>\n

55. Window: [latex]\\left[-0.5,-0.1\\right][\/latex] \u00a0 Range: [latex]\\left[4,\\text{ }100\\right][\/latex]<\/p>\n

\"Graph<\/p>\n

Window: [latex]\\left[0.1,\\text{ }0.5\\right][\/latex] \u00a0 Range: [latex]\\left[4,\\text{ }100\\right][\/latex]<\/p>\n

\"Graph<\/p>\n

57.\u00a0[latex]\\left[0,\\text{ }8\\right][\/latex]<\/p>\n

59.\u00a0Many answers. One function is [latex]f\\left(x\\right)=\\frac{1}{\\sqrt{x - 2}}[\/latex].<\/p>\n

61.\u00a0The domain is [latex]\\left[0,\\text{ }6\\right][\/latex]; it takes 6 seconds for the projectile to leave the ground and return to the ground.<\/p>\n

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