# 2.10 Practice Question Solutions

## Verbal Question Solutions

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 $\text{}x\text{}$ for $\text{}f\left(x\right)=\sqrt[3]{x}\text{}$ because you can take the cube root of any real number. So the domain is all real numbers, $\text{}\left(-\infty ,\infty \right)\text{}$. When dealing with the set of real numbers, you cannot take the square root of negative numbers. So $\text{}x$-values are restricted for $\text{}f\left(x\right)=\sqrt[]{x}\text{}$ to nonnegative numbers and the domain is $\text{}\left[0,\infty \right)$.

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

## Algebraic Question Solutions

7. $\left(-\infty ,\infty \right)$

9. $\left(-\infty ,3\right]$

11. $\left(-\infty ,\infty \right)$

13. $\left(-\infty ,\infty \right)$

15. $\left(-\infty ,-\frac{1}{2}\right)\cup \left(-\frac{1}{2},\infty \right)$

17. $\left(-\infty ,-11\right)\cup \left(-11,2\right)\cup \left(2,\infty \right)$

19. $\left(-\infty ,-3\right)\cup \left(-3,5\right)\cup \left(5,\infty \right)$

21. $\left(-\infty ,5\right)$

23. $\left[6,\infty \right)$

25. $\left(-\infty ,-9\right)\cup \left(-9,9\right)\cup \left(9,\infty \right)$

27. domain: $\text{}\left(2,8\right]\text{}$; range: $\text{}\left[6,8\right)\text{}$

29. domain: $\text{}\left[-4,4\right]\text{}$; range: $\text{}\left[0,2\right]\text{}$

31. domain: $\text{}\left[-5,\text{ }3\right)\text{}$; range: $\text{}\left[0,2\right]$

33. domain: $\text{}\left(-\infty ,1\right]\text{}$; range:$\text{}\left[0,\infty \right)\text{}$

35. domain: $\text{}\left[-6,-\frac{1}{6}\right]\cup \left[\frac{1}{6},6\right]\text{}$; range: $\text{}\left[-6,-\frac{1}{6}\right]\cup \left[\frac{1}{6},6\right]\text{}$

37. domain: $\text{}\left[-3,\text{ }\infty \right)\text{}$; range: $\text{}\left[0,\infty \right)\text{}$

39. domain: $\text{}\left(-\infty ,\infty \right)$

41. domain: $\text{}\left(-\infty ,\infty \right)$

43. domain: $\text{}\left(-\infty ,\infty \right)$

45. domain: $\text{}\left(-\infty ,\infty \right)$

## Numeric Question Solutions

47. $\begin{array}{cccc}f\left(-3\right)=1;& f\left(-2\right)=0;& f\left(-1\right)=0;& f\left(0\right)=0\end{array}$

49. $\begin{array}{cccc}f\left(-1\right)=-4;& f\left(0\right)=6;& f\left(2\right)=20;& f\left(4\right)=34\end{array}$

51. $\begin{array}{cccc}f\left(-1\right)=-5;& f\left(0\right)=3;& f\left(2\right)=3;& f\left(4\right)=16\end{array}$

53. domain: $\text{}\left(-\infty ,1\right)\cup \left(1,\infty \right)$

## Technology Question Solutions

55.

window: $\text{}\left[-0.5,-0.1\right]\text{}$; range: $\text{}\left[4,\text{ }100\right]$

window: $\text{}\left[0.1,\text{ }0.5\right]\text{}$; range: $\text{}\left[4,\text{ }100\right]$

## Extension Question Solutions

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

59. Many answers. One function is $\text{}f\left(x\right)=\frac{1}{\sqrt{x-2}}$.

## Real-World Applications Solutions

60. The domain is  [0, 6];  it takes 6 seconds for the projectile to leave the ground and return to the ground.