hectictar

f(x)  =  2x + 3          and          g(x)  =  3x - 2

f( g( f(2) ) )   =   f( g( 7 ) )   =   f(  19  )   =   41

g( f( g(2) ) )   =   g( f( 4 ) )   =   g( 11 )   =   31

And so...

\(\dfrac{f(\ g(\,f(2)\,)\ )}{g(\ f(\,g(2)\,)\ )}\ =\ \dfrac{41}{31}\)_

We can graph  y = f(x)   by picking  x  values and plotting points.

Notice that the point  (-2, 1)  is NOT on the graph and the point  (-2, -1)  IS on the graph.

Do you understand this so far? Do you see how to finish it? If you're still confused please say so!

area of square   =   ( side length )²

And the side length is the length of the line segment between  (-7, 2)  and  (0, 8)

By the Pythagorean Theorem,

( side length )²   =   ( 0 - -7)²  +  (8 - 2)²

( side length )²   =   7²  +  6²

( side length )²   =   49 + 36

( side length )²   =   85

And again       area of square   =   ( side length )²

We just found that     ( side length )²  =  85     So...

area of square   =   85

Let's say  a  is the odd one and  b  is the even one. Then we can say...

a  =  2b - 7       and       b  =  3a - 9

And...

a + b   =   5 + 6   =   11

We can't find what  x  is equal to, but we can find what  x  has to be greater than.

2( 5x - 9 )  >  13

                            Distribute the  2  to the terms in parenthesees.

10x - 18  >  13

                            Add  18  to both sides of the inequality.

10x  >  31

                            Divide both sides of the inequality by  10 ,  a positive number

x  >  3.1

area of pentagon   =   area of square - area of triangle

area of square   =   (side length)²   =   2²   =   4

To find the area of the triangle, we need to find the coordinates of  A  and  B .

To find point  A , we need to find the y-coordinate of the line when  x = -1 .

\(y\ =\ \frac{x}{2}+1\\~\\y\ =\ \frac{-1}{2}+1\\~\\y\ =\ \frac12\)

So   A  =  (-1, \(\frac12\))

To find point  B , we need to find the x-coordinate of the line when  y = 1 .

\(y\ =\ \frac{x}{2}+1\\~\\ 1\ =\ \frac{x}{2}+1\\~\\ 0\ =\ \frac{x}{2}\\~\\ 0\ =\ x\)

So   B  =  (0, 1)

base of triangle   =   0 - -1   =   1

height of triangle  =  1 - \(\frac12\)   =   \(\frac12\)

area of triangle   =   \(\frac12\)( base )( height )   =   \(\frac12\)( 1 )( \(\frac12\) )   =   \(\frac14\)

Now we can find the area of the pentagon.

area of pentagon   =   area of square - area of triangle   =   4 - \(\frac14\)   =   3.75

\(\ \phantom{=\quad}\dfrac{2^{2010}\:+\:2^{2008}}{2^{2005}\:-\:2^{2003}}\\~\\~\\ =\quad\dfrac{2^{2008}(2^{2}\:+\:1)}{2^{2003}(2^{2}\:-\:1)}\\~\\~\\ =\quad\dfrac{2^{2008}}{2^{2003}}\cdot\dfrac{2^{2}+1}{2^{2}-1}\\~\\~\\ =\quad2^{5}\cdot\dfrac{2^{2}+1}{2^{2}-1}\\~\\~\\ =\quad2^{5}\cdot\dfrac{5}{3}\\~\\~\\ =\quad32\cdot\dfrac{5}{3}\\~\\~\\ =\quad\dfrac{160}{3}\)_

If the sum of the digits of a number is divisible by 9, then that number is divisible by 9.

And if a number is divisible by 9 then it is not prime.

So given that the sum of the digits is 9,  the probability that the number is prime is 0.

Is your equation

this:       7 sin²( x )  -  14 sin( x )  +  2   =   -5

or this:       7 sin( 2x )  -  14 sin( x )  +  2   =   -5

(or something else)

???