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1. x not equal 0

2. x not equal to 1, 2

3. x not equal to -2

4. no restictions

5. x not equal to -1

1.  k(x)  = g(x) + h(x)

2.  k(x)  = g(x) - h(x)

3.  k(x)  = g(x) * h(x)

4.  k(x)  = g(h(x))

5.  k(x)  = g(x) * h(x)

6.  k(x)  = h(g(x))

Assume we want the picture frame to have as close to the Golden Ratio as possible, while maintaining its dimensions in rational numbers.  That would give us (or imply to us) the new dimensions of 15 inches by 25 inches.  This is both your solution and your rationale.

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What is the smallest three-digit number in Pascal's triangle? 

Maybe I'm not understanding the question, because there's only one three-digit number in Pascal's Triangle. 

The third row, to wit:  1-2-1   That's reading horizontally, of course... it would be a stretch to assume otherwise. 

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425 cents/64 oz = 6.64 cents per ounce

Round the number and you get the answer 0.07 or 0.1

Missing some info in your question     Length   Width?

Since we can position the vectors in any convenient manner....Let one vector  = ( 10, 0)    and the other = (0, 5)

This will allow for an easy way to calculate the angle between the resultant and the 10N vector

The resultant  =  √ [10^2 + 5^2 ]  =  √125 =  5√5 N

The angle between the 10N vector and the resultant  =

tan θ = 5/10

arctan (5/10)  = θ ≈ 26.6°

Here is a graph  of both functions along with the solution points  : https://www.desmos.com/calculator/ogpmteqhid

12 % of 7100 = what?

12% x 7100 =   12/100 x 7100 = .12 x 7100 = $ 852     Looks like Amanda is a bit short of funds !  (WHO wants a minivan anyway??)

You are given two points on the line....remember how to find slope?

slope = m = rise/run = (y₁-y_{2) /} (x₁-x₂)     so slope = (10- -2)/(2-8) = 12/-6 = -2

Remember the equation of a line ?    y = m x + b    

    your line will start to look like    y = (-2) x + b        substitue one of the x,y points given to calculate 'b'

                                                    10 = (-2)(2) + b          so b = 14

the equation of the line is then  y = -2x + 14 

b)  do the same methodology for these two given points......let me know what you find ! 

The first two look like normal polynomials to me.

For polynomials,

if the highest power is even then the ends will both be above or both be below the x axis.

if the highest power is odd then one end will be above the x axis and one below.    (both heading off to + or - infinity)

The last one looks like it could be a part of a Sideways parabola.

Now you tell me which is which.   And you tell me why you think so. 

Please no one answer over me.

I have no idea what your teacher is looking for but I'll put my silly 2c in.

These  are

Interval 

rectangle

point 

and Line

 Rectangle does not belong with the others becasue there is no   i    in the spelling.

The point does not belong with the others becasue it only needs a single Letter to name it, all the others need two or more.

The rectangle also does not belong with the others becasue it needs 4 letters to name it, none of the others need that many.

alternatively the expression is trivially factored as

(x+7)(x+1)=0

and you should be able to solve it by inspection from this form.

3.....We can use the same product table with a modification

        1     2     3    4     5    6

1    -2     2     3    4     5     6     

2     2    -8     6    8    10   12

3     3     6   -18  12    15  18

4     4     8   12   -32   20  24

5     5    10  15   20   -50  30

6     6     12  18  24   30   -36

Expected value =

-2(1/36) - 8(1/36) - 18(1/36) - 32(1/36) - 50(1/36) - (36)(1/36) + 2(2/36) + 3(2/36) + 4 (2/36) + 5(2/36) +

6(4/36) + 8(2/36) + 10(2/36) + 12(4/36) + 15(2/36) + 18(2/36) + 20(2/36) + 24(2/36) + 30(2/36) =

[ -2 - 8 - 18 -32 - 50 - 36 + 4 + 6 + 8 + 10 + 24 + 16 + 20 + 48 + 30 + 36 + 40 + 48 + 60 ]

_________________________________________________________________________

                                    36

204

___   =

 36

17/3   ≈   5.66

2.

Greatest value  = 36    Least value  = 1

Possible Products

      1     2     3    4     5    6

1    1     2     3    4     5    6     

2    2     4     6    8    10  12

3    3     6     9  12    15  18

4    4     8   12  16    20  24

5    5    10  15   20   25  30

6    6     12  18  24   30  36

Expected value  =

1(1/36) + 2(2/36) + 3(2/36)  + 4 (3/36) + 5 (2/36) + 6(4/36) + 8(2/36) + 9(1/36) + 10(2/36) + 12(4/36) +

15(2/36) + 16(1/36) + 18(2/36) + 20(2/36) + 24(2/36) + 25(1/36) + 30(2/36) + 36(1/36)  =

[ 1 + 4 + 6 + 12 + 10 + 24 + 16 + 9 + 20 + 48 + 30 + 16 + 36 + 40 + 48 + 25 + 60 + 36 ]

____________________________________________________________________

                                                   36

441 / 36 =

12.25

The sum of the cross sectional area of the  2" pipes must equal the cross sectional area of the larger 10" pipe

Cross sectional area = pi r^2

x pi r^2 = pi 5^2        where x = # of 2" pipes       pi 5^2 = area of 10" pipe

x pi (1)^2 = pi 5^2    divide both sides by pi

x = 5^2

x = 25      Yep, you'll need 25 of the 2" pipes to match the 10" pipe  !

Area of a circle = pi r^2

    pi (3^2) = 9pi     using pi ~ 3.14      9pi = 28.26 cm²

The maximum weight of a bowling ball doesn't matter for this question...maybe that is just a red herring.

We can find the surface area of a sphere only knowing its diameter.

surface area of sphere  =  4 π ( radius )²

radius  =  ( \(\frac12\) )( diameter )  =  ( \(\frac12\) )( 8 + \(\frac12\) )  =  ( \(\frac12\) )( \(\frac{17}{2}\) )  =  \(\frac{17}{4}\)

surface area of sphere  =  4 π ( \(\frac{17}{4}\) )²

surface area of sphere  =  4 π ( \(\frac{289}{16}\) )

surface area of sphere  =  \(\frac{289\pi}{4}\)     (square inches)

1.    

Here's a table  of possible sum outcomes

    1  1   1    6   6    6

1  2   2   2   7   7    7

1  2   2   2   7   7    7

1  2   2   2   7   7    7

6  7   7   7   12 12 12

6  7   7   7  12  12 12

6  7   7   7  12  12 12

Expected value  = 

2(9/36) + 7(18/36) + 12(9/36)  =

2(1/4) + 7(1/2) + 12(1/4)  =

1/2  + 7/2 + 3  =

8/2 + 3 =

4 + 3 =

7

The most likely outcome is 7  and its probability  =  18/36  =  1/2

The sum of the elements in the nth row of Pascal's Triangle  = 2^n

The base 10 log of this is

log 2^n  =  n log 2

So

f(n)               n log 2

____   =      ______    =    n

log 2             log 2

10.) compute the modulus and argument of each complex number.

 a.)   -5

We have the form      -5 + 0i

The modulus is   √ [ (-50^2 + 0^2 ]  =  √25  = 5

The argument is θ  so   tan θ  =   0 / -5  =  pi

b. )  -5 + 5i

Modulus  =  √[ (-5)^2 + (5)^2 ]  = √ [ 50]  = 5√2

The argument is θ  so   tan θ  =  5/-5  = - 1  =  3pi/4 

x  is directly proportional to  y³     so     \(x=ky^3\)     where  k  is a constant.

y  is inversely proportional to  √z     so     \(y=\dfrac{s}{\sqrt{z}}\)     where  s  is a constant.

\(x=ky^3\\~\\ x=k\Big(\frac{s}{\sqrt{z}}\Big)^3 \\~\\ x=\frac{ks^3}{z\sqrt{z}}\)

When   z = 12 ,   x = 3     So...

\(3=\frac{ks^3}{12\sqrt{12}}\\~\\ ks^3= 3\cdot12\sqrt{12}\\~\\ks^3=72\sqrt3\)

When   z = 75 ,

\(x=\frac{ks^3}{z\sqrt{z}}\\~\\ x=\frac{72\sqrt3}{75\sqrt{75}}\\~\\ x=\frac{72\sqrt3}{375\sqrt{3}}\\~\\ x=\frac{24}{125}\)

X = k y^3       y = k/sqrtz      combine to

x = k /sqrt z ^3

or

xsqrt z^3 = k   always

3(sqrt12)^3 = xsqrt( 75) ^3

x = 3 (sqrt12)^3 /(sqrt75)^3

x = 3 *8 /125 = 24/125