CPhill

A function must pass the vertical line test.....that is.....any vertical line drawn on a graph can only cut the graph once

By this criteria...graphs A and D   are functions.....the rest are not

x^2 + 6mx + m  =  0

If this has one real root the discriminant  is  0

So...

(6m)^2 - 4m  =  0

36m^2 - 4m = 0

9m^2 - m  =  0            factor

m (9m - 1)  = 0

Setting each factor to 0 and solving for m produces

m =  0       or     m =  1/9

The equation of the line joining  (0, 0) and (1, 2)  is

y = 2x

So  when y = 1.8   x =  .9

The equation of the line joining (1, 2) and (2, 1) is

y = -(x - 1) + 2

y = -x + 3

So.....when y = 1.8

1.8 = - x + 3

-1.2 = -x

1.2 = x

The equation of the line joining (2, 1) and (3, 3)  is

y = 2(x -3) + 3

y = 2x - 6 + 3

y = 2x - 3

When y = 1.8

1.8 = 2x - 3

4.8 = 2x

2.4 = x

I assume you want the sum of the x coordinates when y = 1.8???

If so...we have

.9 + 1.2 + 2.4   =

4.5

This number of miles is given by :

25100 *  (45 /360)  =  3137.5 milies  ≈   3200 miles  (rounded to the nearest 100 miles)

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  

   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....

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....