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g(x) =  f ( f(x)  =

              1

_________________   =

 2   ( 1 / [2x - 5] ) - 5 

              1

_____________________   =

      2        -       5

______

  2x - 5

       2x  -   5

____________________    =

    2  -  5 [ 2x - 5]

      2x  - 5

____________

    27 - 10x

The x not in the domain  is

27 - 10x  = 0

27 = 10x

27 / 10  

Thank you, everyone!

Thank you so much!

   x + 7                       A                B            

________   =        ______   +  ____

x^2 - x - 2               x - 2           x + 1

This is a partial fraction decomposition

The denominator on the left can be factored as  (x - 2) (x + 1)

Multiply through by this denominator and we get

1x + 7  =   A(x + 1)   +   B(x - 2)

1x + 7  =  (A + B) x  + A - 2B

Equate coefiicients and we get the system

A + B  =  1

A - 2B = 7        subtract the  second from the first

3B = - 6

B = - 2

So A =   A + -2 = 1   ⇒   A =  3

If we connect the three radii of the smaller circles, we will have an equilateral triangle of side 2

And connecting each of these to the center of the largest sphere will create a triangular pyramid with side edges of 3  =  the distance between the centers of a smaller and larger sphere

The distance from any bottom vertex of the triangle to its center is given by :

1 / cos 30°  =   2 / √3   units

The height of this pyramid is given by

√ [3^2  - ( 2/√3)^2]  =  √ [ 9 - 4/3]   =  √[ 23/3] units    

So.....the height of the top of the larger sphere from the  plane is given by

vertical distance from the plane to the center of a smaller sphere + height of the pyramid + radius of larger sphere =

1 + √[23/3] + 2 =

3 + √[23/3]  units  ≈  5.77 units

2 cars travel in opposite directions around a circular path in separate lanes. Car A completes 1 lap in 120 seconds, Car B completes 1 lap in 240 seconds. After the cars pass each other once, how long will it take to pass each other again?

Car A completes 1/2 lap per minute

Car B completes  1/4 lap per minute

So....when they pass each other again, their combined distance traveled = 1 lap

So....we want to solve this where x is the number of minutes until they meet again

x/2 + x/4 =  1

[ 2x + x ]  / 4 = 1

3x / 4  = 1

3x = 4

x = 4/3     minutes   they will pass each other again  = 80 seconds

Thanks, PM, for those resources....

23,328 = 2^5 * 3^6

Will divide the exponents by 3 and add 1 and then multiply them:

5 /3 + 1 =2

6/3  + 1 =3

So, 3 x 2  = 6 perfect cubes.

each step can be denoted as (1,0,0), (0,1,0), (0,0,1)

the sum of the steps equals (2,2,2)

you are not allowed to overshoot in any direction

It should be pretty clear we need to take 2 steps in each direction.

So the total number of paths is just

\(N=\dbinom{6}{2}\dbinom{4}{2} = 90\)

The first term represents choosing two easts say, from the 6 possible steps.

The second term represents choosing two norths from the remaining 4 steps.

Give those the two up steps are fixed.

It doesn't matter which direction you pick for choosing first and second.  The results are the same.

As far as I can see the maximum mean score you can achieve with these conditions is 14.

We have that  

x^2 + (c + 1)x + c  = 0

If this has one real root, it is a double-root and the discriminant  = 0      

So.......

(c + 1)^2  - 4(1)c  = 0

c^2 + 2c + 1 - 4c  =  0

c^2 - 2c + 1  =  0      factor

(c - 1)^2  = 0

So.....c  = 1

This is the only value of c that makes this true....

If we have a y intercept of 8, then the point (0,8) must be on the graph

We have this 

8 = a(0 + 3/2)^2 + 7/4

8 =  a(9/4) + 7/4

32 = 9a + 7

25 = 9a

a = 25/9

So   we have

y =  (25/9)(x + 3/2)^2 + 7/4     

Look at the graph here https://www.desmos.com/calculator/jdr4rgp3vj

Note that none of the given answers are correct......

Not really sure what they want! if they want the arithmetic mean of the 19 numbers...123, 124....etc., then all you have do is add them all up and divide the total by 19. I think it comes to: 5100 / 19 =268.42.

Note : 2.5 ft =  30 in

When the front wheel makes 100 revs, the bike will travel (in inches)

100 *  2pi * 30   =    6000pi   in

So....the number of revs that the back wheel makes is

Total distance traveled  / distance traveled in one rev by back wheel  =

[ 6000 pi ]  / [ 2pi * 4 ]  =    6000/8  =  750 revolutions

Converting feet to inches, we have 2.5*12=30 inches. This is for the front wheel, while the back wheel's radius is 4 inches. Revolutions come hand in hand with the circumference, so we have 2*30*pi=60pi for the front wheel and 2*4*pi=8pi for the back wheel. Now, when 60pi=100 then by indirect proportion \(\frac{60*100}{8}=\boxed{750} \) revolutions. 

thank you

1.  The equation of the asymptote is   y = -2

2.  Look at the graph here :  https://www.desmos.com/calculator/jrasmbb09p

g(x)  is the parent function f(x)  just shifted down 2 units

-2, -1, 0, 0, 1, 2

We have 6C3  =  20 possible sets of outcomes  [order doesn't matter ]

(-2, -1, 0)  (-2, -1, 0) (-2, - 1, 1) ( -2, - 1, 2)

(-2, 0, 0) (-2, 0, 1) (-2, 0, 2)

(-2, 0,1) (-2, 0, 2)

(-2,1,2)

(-1,0,0)  (-1, 0, 1) (-1, 0,2)

(-1,0, 1) (-1, 0, 2)

(-1,1,2)

(0,0,1) (0, 0, 2)

(0, 1, 2)

(0, 1, 2)

The probability  is    4 outcomes / 20  outcomes =    1 / 5

Thanks, Melody, for that resource....!!!

By hand we can do this:

Area between z = 0  and z = 1/2  =   .6915 - .5000 =  .1915

Area between z = 0  and z = -2  =  .5000 - .0228  = .4772

The probability (area) is the sum of these

So.....the probability (area)  is   .1915 + .4772   =  .6687

Thank you. Your answer made perfect sense. I just did not see it. I added all the "arithmetic means and divided by 19 to get 2.16 - was that a next step?

These are the 3-digit numbers that satisfy the given conditions:
[123, 124, 125, 126, 127, 128, 129, 139, 149, 159, 169, 179, 189, 289, 389, 489, 589, 689, 789]
These are the sums of above 3-digit numbers that satisfy the given conditions:
( 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24)
These are the "arithmetic means" of the above sums:
(2, 2.333333333, 2.666666667, 3, 3.333333333, 3.666666667, 4, 4.333333333, 4.666666667, 5, 5.333333333, 5.666666667, 6, 6.333333333, 6.666666667, 7, 7.333333333, 7.666666667, 8)

a) Divide every term by 2.

b) The 5th term is: 10.5 / 2 = 5.25

a) You are already given the common difference of 3. Just subtract 3 from every term.

b) Just multiply every term by 3.

dimensions    4, x+1,  x+11    all in cm        V=924cm^3

so

\(4(x+1)(x+11)=924\\ (x+1)(x+11)=231\\ x^2+12x+11=231\\ x^2+12x-220=0\\ \)

The rest is pretty much already done.    

I don't get what you are asking.