The regular rules of arithmetic apply to function values. Addition and multiplication are commutative, but subtraction and division are not.
The ring operator, signifying a function of a function, is not commutative, either. A simple example can illustrate this:
f(x) = 3
g(x) = x²
(f◦g)(x) = f(g(x)) = f(x²) = 3
(g◦f)(x) = g(f(x)) = g(3) = 3² = 9 . . . . . not equal to 3
The only appropriate choice for Question 19 is ...
A. (f+g)(x) = (g+f)(x)
Answer:
96.4% of people have an IQ score between 69 and 132
Step-by-step explanation:
When the distribution is normal, we use the z-score formula.
In a set with mean
and standard deviation
, the zscore of a measure X is given by:
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
In this question:
![\mu = 100, \sigma = 15](https://tex.z-dn.net/?f=%5Cmu%20%3D%20100%2C%20%5Csigma%20%3D%2015)
What percentage of people have an IQ score between 69 and 132, to the nearest tenth?
The proportion is the pvalue of Z when X = 142 subtracted by the pvalue of Z when X = 69. The percentage is the proportion multiplied by 100. So
X = 132
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![Z = \frac{132 - 100}{15}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B132%20-%20100%7D%7B15%7D)
![Z = 2.13](https://tex.z-dn.net/?f=Z%20%3D%202.13)
has a pvalue of 0.983
X = 69
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![Z = \frac{69 - 100}{15}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B69%20-%20100%7D%7B15%7D)
![Z = -2.07](https://tex.z-dn.net/?f=Z%20%3D%20-2.07)
has a pvalue of 0.019
0.983 - 0.019 = 0.964
0.964*100 = 96.4%
96.4% of people have an IQ score between 69 and 132
From the double-angle identity,
![cos2x=2*sinx*cosx](https://tex.z-dn.net/?f=cos2x%3D2%2Asinx%2Acosx)
we can rewritte our given equation as:
![4sinxcosx-2cosx=0](https://tex.z-dn.net/?f=4sinxcosx-2cosx%3D0)
By factoring 2cosx on the left hand side, we have
![2cosx(2sinx-1)=0](https://tex.z-dn.net/?f=2cosx%282sinx-1%29%3D0)
This equation has 2 solutions when
![\begin{gathered} cosx=0\text{ ...\lparen A\rparen} \\ and \\ 2sinx-1=0\text{ ...\lparen B\rparen} \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20cosx%3D0%5Ctext%7B%20...%5Clparen%20A%5Crparen%7D%20%5C%5C%20and%20%5C%5C%202sinx-1%3D0%5Ctext%7B%20...%5Clparen%20B%5Crparen%7D%20%5Cend%7Bgathered%7D)
From equation (A), we obtain
![x=\frac{\pi}{2}\text{ or }\frac{3\pi}{2}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B%5Cpi%7D%7B2%7D%5Ctext%7B%20or%20%7D%5Cfrac%7B3%5Cpi%7D%7B2%7D)
and from equation (B), we have
![\begin{gathered} sinx=\frac{1}{2} \\ which\text{ gives} \\ x=\frac{\pi}{6}\text{ or }\frac{5\pi}{6} \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20sinx%3D%5Cfrac%7B1%7D%7B2%7D%20%5C%5C%20which%5Ctext%7B%20gives%7D%20%5C%5C%20x%3D%5Cfrac%7B%5Cpi%7D%7B6%7D%5Ctext%7B%20or%20%7D%5Cfrac%7B5%5Cpi%7D%7B6%7D%20%5Cend%7Bgathered%7D)
On the other hand, we can find one more solution from the original equation by substituting x=0, that is,
![\begin{gathered} 2ccos(2\times0)-2cos0=0 \\ which\text{ gives} \\ 2\times1-2\times1=0 \\ so\text{ 0=0} \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%202ccos%282%5Ctimes0%29-2cos0%3D0%20%5C%5C%20which%5Ctext%7B%20gives%7D%20%5C%5C%202%5Ctimes1-2%5Ctimes1%3D0%20%5C%5C%20so%5Ctext%7B%200%3D0%7D%20%5Cend%7Bgathered%7D)
then, x=0 is another solution. In summary, we have obtained the following solutions:
![\begin{gathered} x=0 \\ x=\frac{\pi}{2}\text{or}\frac{3\pi}{2}\text{ and } \\ x=\frac{\pi}{6}\text{or}\frac{5\pi}{6} \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20x%3D0%20%5C%5C%20x%3D%5Cfrac%7B%5Cpi%7D%7B2%7D%5Ctext%7Bor%7D%5Cfrac%7B3%5Cpi%7D%7B2%7D%5Ctext%7B%20and%20%7D%20%5C%5C%20x%3D%5Cfrac%7B%5Cpi%7D%7B6%7D%5Ctext%7Bor%7D%5Cfrac%7B5%5Cpi%7D%7B6%7D%20%5Cend%7Bgathered%7D)
However, the intersection of the last set is empty. So the unique solution is x=0 as we can corroborate on the following picture:
Therefore, the solution set is: {0}