Solutions for Odd-Numbered Section Exercises

1. The domain of a function depends upon what values of the independent variable make the function undefined or imaginary.

3.  There 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].

5. Graph 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.

7. [latex]\left(-\infty ,\infty \right)[/latex]

9. [latex]\left(-\infty ,3\right][/latex]

11. [latex]\left(-\infty ,\infty \right)[/latex]

13. [latex]\left(-\infty ,\infty \right)[/latex]

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

17. [latex]\left(-\infty ,-11\right)\cup \left(-11,2\right)\cup \left(2,\infty \right)[/latex]

19. [latex]\left(-\infty ,-3\right)\cup \left(-3,5\right)\cup \left(5,\infty \right)[/latex]

21. [latex]\left(-\infty ,5\right)[/latex]

23. [latex]\left[6,\infty \right)[/latex]

25. [latex]\left(-\infty ,-9\right)\cup \left(-9,9\right)\cup \left(9,\infty \right)[/latex]

27. Domain: [latex]\left(2,8\right][/latex]   Range [latex]\left[6,8\right)[/latex]

29. Domain: [latex]\left[-4, 4\right][/latex]   Range: [latex]\left[0, 2\right][/latex]

31. Domain: [latex]\left[-5,\text{ }3\right)[/latex]   Range: [latex]\left[0,2\right][/latex]

33. Domain: [latex]\left(-\infty ,1\right][/latex]   Range: [latex]\left[0,\infty \right)[/latex]

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

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

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

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

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

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

47. [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]

49. [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]

51. [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]

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

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

Window: [latex]\left[0.1,\text{ }0.5\right][/latex]   Range: [latex]\left[4,\text{ }100\right][/latex]

57. [latex]\left[0,\text{ }8\right][/latex]

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

61. The 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.