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There are 7 turning points so p must be 8.

Thanks for sharing your solution guest  

Never mind, I got it :).

For anyone who stumbles across this in the future and is curious how to approach it, here was my solution:

Setting \(x = y = 0\), we get \(f(0)f(0) = f(0) + 0\), so \(f(0)^2= f(0)\). Thus, this is only true when \(f(0) = 1\).

To substitute this value into our equation, we can set \(y = -x\). It becomes

 \(\begin{align*} f(x) f(-x) &= f(0) -x^2 \\ &= 1 -x^2 \\ &= (1+x) (1-x). \\ \end{align*}\)

Therefore, \(f(x) = \boxed{1+x} \) or \(\boxed{1-x}\).

If any part is wrong or could've been done more efficiently, please let me know. Hope this helps someone!

x = amt to Xu

4x = amt to Adrian

3(x + 50) = amt to Vicky

2x -100  = amt to Cynthia

5 ( x + 4x)  =  amt to Franklin

4(2x - 100)  =  amt to Vivien

x  + 4x  + 3(x + 50) +  (2x - 100) + 5(x + 4x) + 4(2x - 100)  = 3950   simplify

5x + 3x + 150 + 2x - 100 + 5x  +  20x  + 8x - 400  = 3950

43x  -350   = 3950     add 350 to both sides

43x  = 4300   divide both sides by  43

x  = 100  =   amt to Xu

4(100)=  400 = amt to Adrian 

3(100 + 50) = 450 = amt to Vicky

2(100) - 100 = 100 = amt to Cynthia

5(100 + 4(100) ) = 2500 = amt to Franklin

4 ( 2(100) -100) =  400 = amt to Vivien

Rewrite  the equations as

y= (5/12)x + 45/12       →         y = (5/12)x + 15/4    (1)

y = (5/12)x + 46/12      →         y  = (5/12)x  + 23/6

A perpendicular  line  to  both of  these  will have a slope of  (-12/5)

And if we let this line pass through  (0, 23/6)    we  have  the following equation

y =  (-12/5)x + 23/6

To find the  x intersection of  this line and  (1)  we have

(5/12)x + 15/4 =  (-12/5)x + 23/6

(5/12 + 12/5)x  =  23/6 - 15/4

(169/60)x = 1/12      

x = (1/12) (60/169)  = 5/169

And y = (5/12) (5/169)  + 15/4   =  3815/1014

The distance between these lines  =

sqrt  [ (5/169)^2  +  ( 23/6 - 3815/1014)^2 ]   =  1/13  units

x^2  + y^2   - 4x  -16y  + k  = 0         complete the square on x and y, subtract k from both sides

x^2  -4x  + 4  + y^2 - 16y  + 64   = - k + 4 + 64

(x - 2)^2  + ( y - 8)^2  =  68  - k

The center  =  (2, 8)

The radius  =  8   so    r^2 = 64      so.....k = 4

Here's a graph  :  https://www.desmos.com/calculator/6ucox9q4dw

In circle E,  AB // CD and BC is a diameter.

If m AC = 4x-5 and m BD = 2(x+20), what is m

If m AC = 65 -2x and m BD = 4 (2x-5)-5, what is m

4x-5 = 2(x+20)          ==> x = 22.5    ==> m = 85

 65 -2x = 4 (2x-5)-5        ==> x = 9        ==> m =47

x + 2y  =  0    →  x = -2y

Sub this into  the equation of the circle for x

(-2y)^2  + y^2  -4(-2y)  + 4y   =  0   simplify

4y^2  + y^2  + 8y + 4y   = 0

5y^2  + 12y  = 0      factor

y (5y  + 12) =  0

Set  the factors to 0 and solve

y   = 0                            5y + 12  = 0

x = -2(0)  =0                  5y  = -12

                                        y = -12/5

                                         x=  -2(-12/5)  = 24/5

A = (0,0)    B  =( 24/5, -12/5)

AB =  sqrt [  (24/5)^2  + (-12/5)^2  ]  =    (1/5)  sqrt  [ 576 + 144 ] =  (1/5) sqrt (720)  =  (12/5)*sqrt (5)  units

See the graph here : https://www.desmos.com/calculator/chrvtjmvty

2x - y + 4  = 0 

This boils down to finding  the  distance  between the origin and  this line

This is given by

l  2(0)  - 0  + 4 l                   4            

_____________    =        ______   =  the radius of the circle  

sqrt  (2^2 + 1^2)               sqrt (5)

Area  of circle  =    pi * radius^2  =  pi *(4/sqrt(5) )^2 =   (16/5)pi  units^2

Here's the graph to show this  :  https://www.desmos.com/calculator/xlkqr2a40h

https://web2.0calc.com/questions/circle-gamma-is-the-incircle-of-triangle-abc-and_2

Thx, Asinus....here's another way

Distance  between  (3,4)  and  line  4x + 8y  + 7  = 0

l 4(3)  + 8(4)  + 7 l                   51               51                51 sqrt (5)

________________ =         ______  =  ________  =  _________  ≈   5.70 units

sqrt ( 4^2  + 8^2)                 sqrt (80)      4sqrt (5)              20

LOL!!!!....no big  secret....it's  Chris  (I have this on good authority  !!!)

When the  x  values are the same.....the slope is undefined....to see why this is so......

slope  = [ 9 - 1 ]  / [ 1 - 1 ] =    8 / 0

But  we can't divide  by 0, so this slope is undefined  !!!

The length of segment OS ≈ 11.504 units

This will have no real solutions when  the discriminant  is  <  0 

So  

(3)^2  - 4(m) (1 - m)   <  0

9  - 4m + 4m^2   <  0

4m^2  - 4m  + 9  <  0

Looking at the graph here and subbing x  for m,note that this function is  always >  0

https://www.desmos.com/calculator/q6jmvmvy7y

So....no values of m  give non-real solutions

The length of the  diagonal  on the base of the  pyramid = 8sqrt (2) * sqrt (2)   =16

Hafl of this length  = 8

Height of the  pyramild =  sqrt  (10^2 - 8^2 ) = sqrt 36  = 6

Then the height of the  smaller pyramid =  6 - 4   = 2

These pyramids  are  similar  so the scale factor  is  2/6 =  1/3

Volume of the larger pyramid  =   (1/3) side^2  * height  =  (1/3) (8sqrt (2) )^2 * 6 = 2 * 128=  256 units^3

Volume of smaller pyramid =  volume of large pyramid * scale factor ^3  =  256 (1/3)^3 = 256/8 = 32 units^3

Thank you CPhill! 😊

You're very welcome! Glad to help.😃

Not quite

Joy bought X

Ansen bught 2X

so far so good.

Kingley bought 3 less than 3 times as many slices as joy.

What is 3 times as many slices as Joy?   what is 3 less than that?

Jacky bought twice the sum of what joy bought and 5. 

This is badly worded but it means

But it means. 

add 5 to what Joy bought, and THEN you double it.   That is what Jacky gets.

The last one is similar.

NOW   try again.

Thanks, Melody  !!!!

Good job, Spring   !!!

No prob  !!!