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avatar+1442 

1: The function h(x) is defined as: 

 

        { (floor(4x)) if x < / = pi

h(x)={ 3-x            if pi < x

        { x^2           if 5.2 < x

 

Find h(h(sqrt2))

 

2: Find constants A and B such that

 

          (x+7)/(x^2-x-2) = A/(x-2) + B/(x+1)

 

for all x such that x does not equal -1 and x does not equal 2. Give your answer as the ordered pair (A,B).

 

3: Suppose that  | a - b | + | b - c | + | c - a | = 20. What is the maximum possible value of | a - b |?

 

 

Thanks so much! Feel free to answer individually!

AnonymousConfusedGuy  Dec 15, 2017
edited by AnonymousConfusedGuy  Dec 15, 2017
edited by AnonymousConfusedGuy  Dec 15, 2017
 #1
avatar+91097 
+1

2: Find constants A and B such that

 

          (x+7)/(x^2-x-2) = A/(x-2) + B/(x+1)

 

We can use partial fractions here

 

(x +7)  / [ ( x -2) (x + 1) ]   =    [ A (x + 1)  + B(x - 2) ] / [ ( x - 2) (x+1)]

 

And we can solve this

 

x + 7  =  A(x + 1)  + B(x - 2)   simplify

 

x + 7  = Ax + A  +  Bx  - 2B     

 

x + 7  =  (A + B)x  + ( A - 2B)      equate coefficients

 

A + B  =  1

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

 

3B  = -6

B = -2

 

So   A  + (-2)  =    1       ⇒  A  =  3

 

So    { A, B}  =  { 3, -2}

 

 

cool cool cool

CPhill  Dec 15, 2017
 #2
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+2

Since the differences between the three variables are "absolute", then the maximum difference between a and b must be 10!!. You may assign any two values to them as long as they are separated by 10. And the value of c would be the average of the two. So, if we pick:

a =110, b=100 and c=105, then: | 110 - 100 | + | 100 - 105 | + | 105 - 110 | = 20

                                                                    10         +          5            +        5             = 20

 

So the maximum of: a - b =10

Guest Dec 15, 2017
 #3
avatar+1442 
+3

Thanks for both your answers!

AnonymousConfusedGuy  Dec 15, 2017
 #4
avatar+91097 
+1

OK......ACG

 

In your first  problem, you  have

 

floor (4x)   if   x

 

If x   is what  ???

Also......if  x > pi   ......  either of the other two functions might apply

 

Did you leave something out  ???

 

 

cool cool cool

CPhill  Dec 15, 2017
 #5
avatar+1442 
+2

Oh oops, I did! Fixed now!

AnonymousConfusedGuy  Dec 15, 2017
 #6
avatar+91097 
+2

1.

 

h (h(sqrt 2) )

 

Since   sqrt 2 ≤  pi.......use the first function

 

So   floor  (4 * sqrt2)  =  floor ( ≈ 5.6)  =  5

 

So

 

h (h (sqrt 2))  =  h (5) 

 

Since  5     is     > pi   but <  5.2

 

We use the second function, 3 - x

 

So

 

h (5)  =   3  -  5     =     - 2

 

 

 

cool cool cool

CPhill  Dec 15, 2017
edited by CPhill  Dec 15, 2017
edited by CPhill  Dec 15, 2017

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