Two events are said to be mutually exclusive (or disjoint) if it is impossible for both of them to occur on the same outcome. Which of the following pairs of events are mutually exclusive? a. You roll a sum of 7 with a pair of dice; you get doubles on the same roll. b. You roll a sum of 8 with a pair of dice; you get doubles on the same roll. c. Isaac wears white shoes today to class; Isaac wears black shoes today to class. d. Yen owns white shoes; Yen owns black shoes. e. Pat is a boy, Pat said most students in his school are friendly. f. Bernardo said that all students are friendly; Bernardo said most students are friendly.

Explain why the fixed points for a function f(x) correspond to the points of intersection of the graphs of y = f(x) and y = x.

For each of the following linearly dependent sets, find a redundant vector v in S and verify that $\operatorname{span}(S-\{\mathbf{v}\})=\operatorname{span}(S)$. (a) $S=\{[4,-2,6,1],[1,0,-1,2],[0,0,0,0],[6,-2,5,5]\}$, *(b) $S=\{[1,1,0,0],[1,1,1,0],[0,0,-6,0]\}$, (c) $S=\left\{\left[x_{1}, x_{2}, x_{3}, x_{4}\right] \in \mathbb{R}^{4} | x_{i}=\pm 1\right.$, for each $i \}$.

The following topics are appropriate for either individual or group research projects. Use the Internet to investigate each topic. Federal Tax Procedures Questioned over Issues of Fairness (Examples include the marriage penalty, the alternative minimum tax (AMT), and capital gains rates.)

Find the intervals on which $f$ increases and the intervals on which $f$ decreases.

$$f(x)=|x^2-5|$$

A function $y = f(x)$ is transformed to the function $y = -f(-x)=g(x):$ a. Describe the nature of the transformation. b. If (3, -7) lies on $y = f(x),$ find the transformed point on g(x). c Find the point on $f(x)$ that transforms to the point (-5, -1).

Aspirin is an acidic substance. Describe the pathway by which an aspirin overdose causes an increase in respiratory rate, and explain how this increase would be helpful to the victum.

Determine the simple moving averages (SMA) by subtraction and addition for each set of 10 consecutive day dosing prices. 4-day SMA $26.31,$25.94, $27.65,$27.13, $26.81,$26.65, $26.55,$25.89, $25.82,$25.87.

Change each exponential statement to an equivalent statement involving a logarithm. $9=3^{2}$

Identify a pattern in each list of numbers. Then use this pattern to find the next number. (More than one pattern might exist, so it is possible that there is more than one correct answer.) $1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}$, ____

Find the length of the line segment connecting each pair of points. a. (1, 3), (6, 3) b. (-4, 2), (1, 2) c. (7, -5), (7, -3) d. (-8, -4), (8, -4)

Simplify each expression. $(p^5)^6$

The sample size needed to estimate the difference between two population proportions to within a margin of error E with a confidence level of $1-\alpha$ can be found by using the fo llowing expression:

$$E = z _ { \alpha / 2 } \sqrt { \frac { p _ { 1 } q _ { 1 } } { n _ { 1 } } + \frac { p _ { 2 } q _ { 2 } } { n _ { 2 } } }$$

Replace $n _ { 1 }$ and $n _ { 2 }$ by $n$ in the preceding formula (assuming that both samples have the same size) and replace each of $p _ { 1 } , q _ { 1 } , p _ { 2 } ,$ and $q _ { 2 }$ by 0.5 (because their values are not known). Solving for $n$ results in this expression:

$$n = \frac { z _ { \alpha / 2 } ^ { 2 } } { 2 E ^ { 2 } }$$

Use this expression to find the size of each sample if you want to estimate the difference between the proportions of men and women who own smartphones. Assume that you want 95% confidence that your en-or is no more than 0.03.

Perform the indicated operations, expressing answers in simplest form with rationalized denominators.$\frac{\sqrt{T^{4}-V^{4}}}{\sqrt{V^{-2}-T^{-2}}}$