CPhill

Here's 3

5         4           - 1           -10             -23              -40

    - 1            -5         - 9             -13              -17

             -4            -4          -4              -4

Since the second differences are all the same, this is a quadratic

To find the polynomial, we have this system :

a     +   b   +   c  =  5

4a   + 2b   +  c  =   4

9a  + 3b   +   c   =   -1

Subtract   the first equation from the second and the second equation from the third and we have

3a + b  = -1

5a  + b  = -5      subtract the first equation from the second and we have

2a  = -4

a  = -2

And using     3a + b  = -1   to find b, we have.......3(-2) + b  = -1  →    b  = 5

And using a + b + c  = 5 to find c we have   -2 + 5 + c  = 5    →  c  = 2

So the generating  polynomial  is :     -2x^2 + 5x + 2

Here's 4

20                  4                      0                   20                         76                      180

        -16                    -4                   20                       56                   104

                     12                    24                   36                      48

                                12                     12                     12

Since the third differences are the same, this is a cubic polynomial

And to find the generating polynomal, we have this system

a   +    b   +   c    +  d    =   20

8a  +  4b  +  2c   +  d   =     4

27a + 9b  +  3c  +  d    =     0

64a + 16b  + 4c   + d   =    20

Solving this system in a similar manner to (3)   we have that  a = 2, b = -6, c = -12, d = 36

So the generating polynomial  is        2x^3  - 6x^2  -12x  + 36

Perimeter of a rectangle  = 2 [ Base + Height ]  .....so...

88  =  2 [ (x + 4) + ( x - 6) ]      divide both sides by 2 and simplify within the parentheses

44 = 2x - 2        add 2 to both sides

46  = 2x             divide both sides by 2

x  = 23

So.....  the base  = ( x + 4)  = ( 23 + 4)    =   27

And the height   =  ( x - 6)  =   ( 23 - 6 )  =  17

If you're referring to an inverse function.....the original function pairs some x with some y

The inverse reverses these coordinates

Example

Original function       f(x)  =   3x + 5

Inverse function      f^-1(x) = [ x - 5] / 3

Note that  if  (0, 5)   is a point on the original function.......then  note that ( 5, 0)  is on the inverse function

2.75  =

2 + 75/100  =

2 +    [ 3 * 25 ]  / [ 4 * 25 ]         cancel 25's

2 +  3 / 4  =

11 / 4

48/2(9+3)

Trincent, we want to simplify within the parentheses, first.....so we have

48/2 (12)

Now.....we perform multiplications/divisions as we encounter them from left to right.....so......divide 48 by 2

24 (12)   .....now......multiply 24 times 12  = 288

We  want to use these two identities

cos (x)  = cos (-x)    and      csc (-x)  = -csc (x)

So

cos (-105°)  =  cos (105°)  = -0.26

And

csc ( -105°)   =  -csc (105°) = -1.03  .....so..... - [- csc (105°) ] = csc (105°) =  - [ -1.03]  =  1.03

If f(x) = 4x^2 - x+1, then f(a+1) equals

We are just  substituting (a + 1)  for x.....so we have

4(a + 1)^2 - (a + 1) + 1       simplify

4 ( a^2 + 2a + 1) - a - 1 + 1

4a^2 + 8a + 4  - a - 1 + 1

4a^2 + 7a + 4

15  / 60    = 1 /4

Let x be the percentage.......and we have that......

1 /4   =   x  /100            multiply both sides by 100

100 / 4  =  x   =   25   (percent)

Triangle PRS is equilateral.....so angle PSR  = 60° = angle PSQ 

And PS  = 2   and  QS  = 4

And by the Law of Cosines we have that

PQ  =  sqrt ( 2*2 + 4^2  - 2(4)(2) cos 60 )  =  sqrt ( 20 - 8)  = sqrt (12) = 2sqrt(3)

And by the Law of Sines, we have

sin PSQ / PQ  =  sin PQS / PS

sin 60 / 2sqrt (3)   = sin PQS / 2

(sqrt (3) / 2) / [ 2sqrt (3) ]  =  sin PQS / 2

(1/4)  =  sin PQS / 2

sin PQS  =  1/2  ...    thus PQS  = 30°

So..since PS is the shortest side in triangle PQS, then angle PQS is the smallest angle   = 30°