CPhill

3.  f(x)  = 3x  + 2

Here's the way I learned it..... for  f(x), write "y".... get x by itself... "swap" x and y

 and for "y"  write f^-1(x)  [this notation represents the inverse ]

So we have

y  = 3x+ 2

y - 2  = 3x

[y - 2] / 3  = x        "swap"  x and y

[x - 2 ] / 3  = y

[x -2 ] / 3  = f^-1(x)

BTW....swapping x and y like you did in your first step is  OK, too.....!!

#4   looks good...!!!

Here's  the last two

f(x)  = x^2  + 8

y = x^2 + 8

y - 8  = x^2        take both roots

±√ [y -8] = x      swap  x and y

±√[ x - 8 ]  = y

±√[x - 8]  =  f^-1(x)

f(x)  = (x + 3)^2

y = (x + 3)^2      take both roots

±√y = x + 3

±√y  - 3   = x      swap x and y

±√x  - 3   = y

±√x  - 3  =  f^-1(x)

Nice, tertre....!!!!!!

How many of the first 100 rows contain at least one even entry and no odd entries other than 1  ??

I'm not sure, but looking at the patterns in the triangle for the first 20 rows shows that evey 2ⁿ row [ for  n ≥ 1 ]  will only contain even entries  [except for the beginning and ending  "1" entries in these rows]

So....we should have  6   rows since   2⁶  = 64th row.....   but 2⁷   = 128th row

To  find the inverse  just  reverse the co-ordinates

So...for the first one, we have

(-3, 1), (3, - 2), (1, 5),  ( 4, 6)

It's a function as long as no two different y values are paired with the same x value

[ In other words...an x value cannot be paired with two different y's ]

This doesn't happen, so the inverse is a function

For the second, we have

(7, - 5), (-8, -6), (-2,1) , (3, 10)

No two different y's are paired with the same x value, so this is also a function

There  are  [ 15 * 16 ] / 2 =  15 * 8   = 120  elements in the first  15 rows of Pascal's Triangle

And there  are   14 * 2  + 1 = 29  "1s"  -   two "1s" on every row except for one on Row  "0"

So....the probability of choosing a "1"  is   29 / 120

Oh....I'm  often incorrect....but....I try  NOT to be  !!!

Well....two great minds couldn't possibly be incorrect, could they   ???

LOL!!!

The  value of the element is    (what  ??? )

h(t)  =  -16t^2  + 120t   +  6

The time  to reach the max height   =    -120  /[ 2 * -16 ]  = -120 / -32  = 3.75 sec

And....to find the max height, plug this value back into the function  =

-16(3.75)^2 + 120 (3.75)  +   6  =   231 ft

Note that the  change in height is [231 - 6 ]  = 225 ft

So...the average rate of change  in height  from intial height to max height   =   [231- 6] / 3.75  =

225 / 3.75  = 60 ft / sec

Every person    shakes  hands with  10  other people

So...  12 * 10   =  120

But handshake  A - C   is the same as handshake  C  - A

So...we need to  divide this by 2

So  120 / 2   = 60  handshakes