CPhill

For the first function the range is  [-3, inf )

For the second function the range is [3 , inf)

So..taking the  greater range   →    [ -3, inf)

 Rearrange as    ...     y =  8 - abs ( x  -1)

This will form a triangle  

The max height occurs when x = 1   and the  height  =   8

To find the x intercepts we have

8 - abs (x -1)  = 0

8 =  abs ( x -1)........this sets up  two equations

8 =  x - 1             - 8  =  x  -1

x = 9                     x =  -7

So the x intercepts are   -7  and 9   and the base of the  triangle =   9 -  (- 7 )  = 16

So.....the area =   (1/2) (16)(8)  = 64

f(12) =  12/2 + 1 =  7

f(f(12))  = f(7)  = 3(7)  -1 =  20

f(f(f(12))) = f(20) = 20/2 + 1 = 11

f(f(f(f(12)))) = f (11) = 3(11) - 1 =   32

Very nice, Melody    !!!!!

m         n

4          3                 x^2 - 3x + 2      roots = ( 1, 2)

12        5                 x^2 - 5x + 6      roots = ( 2,3)

24        7                 x^2 - 7x + 12    roots ( 3,4)

40        9                 x^2  - 9x + 20   roots ( 4 , 5)

.

.

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3960   99                x^2 - 99x + 1980 roots (44,45)   

Number of n  =   (99 - 3) / 2  + 1  =  48 + 1 =    49   

See the graphs here :  https://www.desmos.com/calculator/kuekhvapfp

They never intersect........so there is no region to be determined

We have

(x - 2)^2 + ( y + 1)^2 - 4^2  = (x -2)^2 + ( y - 5)^2  - (sqrt 10)^2

(y + 1)^2  - 16 =  (y -5)^2  - 10

(y + 1)^2  - (y -5)^2  = 6

y^2 + 2y + 1  - ( y^2 - 10y + 25)  = 6

12y - 24 = 6

12y = 30

y = (30/12)  = (5/2)  = 2.5

And

(x -2)^2  + (y + 1)^2 = 16

(x -2)^2  + ( 2.5 + 1)^2  = 16

( x - 2)^2  + 12.25   = 16

(x -2)^2  = 16 - 12.25

(x - 2)^2  = 3.75

x - 2 =  sqrt (3.75)             x -2  = -sqrt (3.75)

x =  2 + sqrt (3.75)  =  x coord of A          x = 2 - sqrt (3.75)  =  x coord of   B

AB =   ( 2 + sqrt (3.75) - ( 2 -sqrt (3.75) ) =  (2sqrt (3.75))

(AB)^2 =  (2sqrt (3.75))^2 =   4 (3.75)  =   15

The slope of the line =  3

The equation of the line is

f(x)   = 3x - 5

f ( f(x))  =   3 (3x  -5) - 5

So

3( 3x  -5) - 5  =  3

9x -15 =  8

9x = 23

x = 23 / 9

Proof

f (23/9)  =  3 (23/9) - 5   = 8/3

f ( 8/3) =  3 (8/3) - 5   =  3

a + b = 10     →  b =  10 - a    (1)

a^4 + b^4 =  2800       (2)

Sub (1)   into (2) and we have that

a^4  + ( 10 - a)^4 =  2800

a^4 + a^4 - 40a^3 + 600a^2 -4000a + 10000 =  2800

2a^4  - 40a^3 + 600a^2 - 4000a - 7200 =  0

a^4 - 20a^3  + 300a^2 - 2000a - 3600  = 0

The  (real) solutions to this are

a ≈ 11.451   b ≈ -1.451

or

a  ≈ -1.451   b ≈ 11.451

Very , very nice BuilderBoi.....this one had me stumped   !!!!

{Drawing B'F  was the key .....very crafty !!! )

I'd give you 5 points if  I could !!!