Since csc(x) is the reciprocal identity of sin(x), they both have the same period, and since sin(x) has a period of 2π, so does csc(x).
<h2>Answer </h2>
negative
<h2>Explanation </h2>
Let's find the slope of our line using the slope formula:
![m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}](https://tex.z-dn.net/?f=m%3D%5Cfrac%7By_%7B2%7D-y_%7B1%7D%7D%7Bx_%7B2%7D-x_%7B1%7D%7D)
where
is the slope of the line
are the coordinates of the first point on the line
are the coordinates of the second point
From the graph we can get the points (-2, 1) and (2, -1), so
,
,
, and
. Let's replace the values in our formula to find
:
![m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}](https://tex.z-dn.net/?f=m%3D%5Cfrac%7By_%7B2%7D-y_%7B1%7D%7D%7Bx_%7B2%7D-x_%7B1%7D%7D)
![m=\frac{-1-1}{2-(-2)}](https://tex.z-dn.net/?f=m%3D%5Cfrac%7B-1-1%7D%7B2-%28-2%29%7D)
![m=\frac{-2}{2+2}](https://tex.z-dn.net/?f=m%3D%5Cfrac%7B-2%7D%7B2%2B2%7D)
![m=\frac{-2}{4}](https://tex.z-dn.net/?f=m%3D%5Cfrac%7B-2%7D%7B4%7D)
![m=-\frac{2}{4}](https://tex.z-dn.net/?f=m%3D-%5Cfrac%7B2%7D%7B4%7D)
![m=-\frac{1}{2}](https://tex.z-dn.net/?f=m%3D-%5Cfrac%7B1%7D%7B2%7D)
The slope of the graph of the linear function is
; since
is a negative number, the slope of the graph is negative.
Answer:
b. Jason pays $ 3 on the $ 3 he owes in fines.
Step-by-step explanation:
an amount owing is represented as negative. the action of paying dues or making a payment is treated as positive.
therefore, Jason pays +3 to the owed 3. this, +3 -3
Answer:
b. n=5
Step-by-step explanation:
Because 30 divided by 36 = 0.83333
And 180 divided by 6 = 30 so that eliminates a.
Then if you plug in 5 for n 5 divided by 6 = .833333
Therefore the answers match and n = 5
Hope this helps.
![{\large\underline{\sf{Solution-}}}](https://tex.z-dn.net/?f=%20%7B%5Clarge%5Cunderline%7B%5Csf%7BSolution-%7D%7D%7D)
Given function is
![\rm \longmapsto\:y = \sqrt{ {x}^{2} - 4}](https://tex.z-dn.net/?f=%5Crm%20%5Clongmapsto%5C%3Ay%20%3D%20%20%5Csqrt%7B%20%7Bx%7D%5E%7B2%7D%20%20-%204%7D%20)
We know
Domain of a function is defined as set of those real values of x for which function is well defined.
So, y is defined when
![\rm \longmapsto\: {x}^{2} - 4 \geqslant 0](https://tex.z-dn.net/?f=%5Crm%20%5Clongmapsto%5C%3A%20%7Bx%7D%5E%7B2%7D%20-%204%20%5Cgeqslant%200)
![\rm \longmapsto\: {x}^{2} - {2}^{2} \geqslant 0](https://tex.z-dn.net/?f=%5Crm%20%5Clongmapsto%5C%3A%20%7Bx%7D%5E%7B2%7D%20-%20%20%7B2%7D%5E%7B2%7D%20%20%5Cgeqslant%200)
![\rm \longmapsto\: (x - 2)(x + 2) \geqslant 0](https://tex.z-dn.net/?f=%5Crm%20%5Clongmapsto%5C%3A%20%28x%20-%202%29%28x%20%2B%202%29%20%20%5Cgeqslant%200)
![\rm\implies \:x \leqslant - 2 \: \: or \: \: x \geqslant 2](https://tex.z-dn.net/?f=%5Crm%5Cimplies%20%5C%3Ax%20%5Cleqslant%20%20-%202%20%5C%3A%20%20%5C%3A%20or%20%5C%3A%20%20%5C%3A%20x%20%5Cgeqslant%202)
![\rm\implies \:x \: \in \: ( - \infty , \: - 2] \: \cup \: [2, \: \infty )](https://tex.z-dn.net/?f=%5Crm%5Cimplies%20%5C%3Ax%20%5C%3A%20%20%5Cin%20%5C%3A%20%28%20-%20%20%5Cinfty%20%2C%20%20%5C%3A%20-%202%5D%20%5C%3A%20%20%5Ccup%20%5C%3A%20%5B2%2C%20%5C%3A%20%20%5Cinfty%20%29)
Now, <u>To find the range of function </u>
We know,
Range of a function is defined as set of those real values which is obtained by assigning the values to x.
So,
![\rm \longmapsto\:y = \sqrt{ {x}^{2} - 4}](https://tex.z-dn.net/?f=%5Crm%20%5Clongmapsto%5C%3Ay%20%3D%20%20%5Csqrt%7B%20%7Bx%7D%5E%7B2%7D%20%20-%204%7D%20)
On squaring both sides, we get
![\rm \longmapsto\: {y}^{2} = {x}^{2} - 4](https://tex.z-dn.net/?f=%5Crm%20%5Clongmapsto%5C%3A%20%7By%7D%5E%7B2%7D%20%3D%20%20%7Bx%7D%5E%7B2%7D%20-%204)
![\rm \longmapsto\: {y}^{2} + 4= {x}^{2}](https://tex.z-dn.net/?f=%5Crm%20%5Clongmapsto%5C%3A%20%7By%7D%5E%7B2%7D%20%20%2B%204%3D%20%20%7Bx%7D%5E%7B2%7D)
![\rm \longmapsto\:x = \sqrt{ {y}^{2} + 4 }](https://tex.z-dn.net/?f=%5Crm%20%5Clongmapsto%5C%3Ax%20%3D%20%20%5Csqrt%7B%20%7By%7D%5E%7B2%7D%20%2B%204%20%7D%20)
Now, x is defined when
![\rm \longmapsto\: {y}^{2} + 4 \geqslant 0 \: which \: is \: always \: true \: \forall \: y \in \: real \: number](https://tex.z-dn.net/?f=%5Crm%20%5Clongmapsto%5C%3A%20%7By%7D%5E%7B2%7D%20%2B%204%20%5Cgeqslant%200%20%5C%3A%20which%20%5C%3A%20is%20%5C%3A%20always%20%5C%3A%20true%20%5C%3A%20%20%5Cforall%20%5C%3A%20y%20%5Cin%20%5C%3A%20real%20%5C%3A%20number)
But,
![\rm \longmapsto\:y \geqslant 0](https://tex.z-dn.net/?f=%5Crm%20%5Clongmapsto%5C%3Ay%20%5Cgeqslant%200)
<u>Hence, </u>
![\rm\implies \:Range \: of \: function \: = [0, \: \infty )](https://tex.z-dn.net/?f=%5Crm%5Cimplies%20%5C%3ARange%20%5C%3A%20of%20%5C%3A%20function%20%5C%3A%20%20%3D%20%5B0%2C%20%5C%3A%20%20%5Cinfty%20%29)
So, we have
![\rm\implies \:Domain \: of \: function : x \: \in \: ( - \infty , \: - 2] \: \cup \: [2, \: \infty )](https://tex.z-dn.net/?f=%5Crm%5Cimplies%20%5C%3ADomain%20%5C%3A%20of%20%5C%3A%20function%20%3A%20x%20%5C%3A%20%20%5Cin%20%5C%3A%20%28%20-%20%20%5Cinfty%20%2C%20%20%5C%3A%20-%202%5D%20%5C%3A%20%20%5Ccup%20%5C%3A%20%5B2%2C%20%5C%3A%20%20%5Cinfty%20%29)
and
![\rm\implies \:Range \: of \: function \: = [0, \: \infty )](https://tex.z-dn.net/?f=%5Crm%5Cimplies%20%5C%3ARange%20%5C%3A%20of%20%5C%3A%20function%20%5C%3A%20%20%3D%20%5B0%2C%20%5C%3A%20%20%5Cinfty%20%29)