1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
bazaltina [42]
3 years ago
11

Let $f(x)$ be a function defined for all positive real numbers satisfying the conditions $f(x) > 0$ for all $x > 0$ and $f

(x - y) = \sqrt{f(xy) + 1}$ for all $x > y > 0$. Determine $f(2009)$.
Mathematics
2 answers:
algol [13]3 years ago
3 0
Suppose we choose x=1 and y=\dfrac12. Then

f(x-y)=\sqrt{f(xy)+1}\implies f\left(\dfrac12\right)=\sqrt{f\left(\dfrac12\right)+1}\implies f\left(\dfrac12\right)=\dfrac{1+\sqrt5}2


Now suppose we choose x,y such that

\begin{cases}x-y=\dfrac12\\\\xy=2009\end{cases}


where we pick the solution for this system such that x>y>0. Then we find

\dfrac{1+\sqrt5}2=\sqrt{f(2009)+1}\implies f(2009)=\dfrac{1+\sqrt5}2

Note that you can always find a solution to the system above that satisfies x>y>0 as long as x>\dfrac12. What this means is that you can always find the value of f(x) as a (constant) function of f\left(\dfrac12\right).
Crazy boy [7]3 years ago
3 0

Solution:

 It is given that, f(x) is a function such that, defined for all positive real numbers satisfying the conditions ,f(x) > 0 ,for all x > 0 , and also

    f(x-y)=\sqrt{f(xy)+1}\\\\x>0,y>0\\\\for, x=1, \text{and} y=\frac{1}{2}\\\\f(1-\frac{1}{2})=\sqrt{f(1\times \frac{1}{2})+1}\\\\f(\frac{1}{2})^2=f(\frac{1}{2})+1\\\\f(\frac{1}{2})=\frac{1+\sqrt{5}}{2}

Now, suppose

x=2009, y=0

f(2009-0)=\sqrt{f(2009*0)+1}\\\\f(2009)=\sqrt{f(0)+1}\\\\f(2009)=\sqrt{f(\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}})+1}\\\\f(2009)=\sqrt{\sqrt{f(\frac{1}{\sqrt{2}}*\frac{1}{\sqrt{2}})+1}+1}\\\\f(2009)=\sqrt{\sqrt{f(\frac{1}{2})+1}+1}\\\\f(2009)=\sqrt{\sqrt{\frac{1+\sqrt{5}}{2}+1}+1}\\\\f(2009)=\sqrt\sqrt{\frac{3+\sqrt{5}}{2}}+1}\\\\f(2009)=\sqrt \sqrt{{5.236}{2}}+1}\\\\=\sqrt{3.6180}\\\\=1.9021

You might be interested in
Vector has 4 orange picks for every 3 green picks. If 8 out of the picks are orange, how many picks are green
Natali [406]

Answer: 32 :24

Step-by-step explanation:

5 0
3 years ago
Is this statement true or false?<br> If sin 30º = 0.5, then sin -- 0.5 = 30°.<br> true<br> false
masya89 [10]

Answer:

true

Step-by-step explanation:

5 0
3 years ago
Can someone help me with this
Brrunno [24]

Answer:

Step-by-step explanation:

8.) For a triangles sides to make sense, you must be able to add up two values of the triangle, and the result should be more than the third side. Add the lowest values and see if the result is greater than the biggest number:

4.2+7.9=12.1

12.1 is less than the given side, 13.3, so a triangle cannot have the lengths.

10.) 6<x<22

To find the range for the third side of the triangle, you need to find how small x can be (the missing side) and you need to see how large it can be.

You need to see how small it can be because any two sides have to be greater than the third side. You also need to see how big it can be because, if it's too big, the other two sides will be less than the third side, which would make an open shape (see picture).

To find the range, first see how small. Subtract the known sides:

22-6=16

So, x has to be greater than 16.

x > 16

Now add the known sides:

22+6=28

x needs to be less than 28 for the other two sides to be greater than x:

x < 28

Insert the inequalities into a single inequality:

16 < x <28

X has to be greater than x, but less than 28.

4 0
3 years ago
Read 2 more answers
Jim writes 10^4 words every day how many words does he writes in 10^3 days?​
Alchen [17]
1,000 words

10^4=10,000
10^3=1,000
7 0
3 years ago
What is the slope of the line that contains the points (9,-4) and (1, -5)
vlada-n [284]

Answer:

A. 1/8

Step-by-step explanation:

m = rise/run

m = (y2-y1)/(x2-x1)

m = (-5- - 4)/(1-9)

m = (-5+4)/(-8)

m = (-1)/(-8)

m = 1/8

5 0
3 years ago
Read 2 more answers
Other questions:
  • What number is 60% of 65?
    11·1 answer
  • Six times the sum of a number x and 4.
    14·1 answer
  • A triangle’s height is 6c meters and the base is c – 1 meters. Brian knows that the area of the triangle is 18 square meters and
    14·2 answers
  • 11 m
    7·1 answer
  • Pls help me out i need it​
    8·1 answer
  • Evaluate the expression | 9 | + | - 2|
    10·2 answers
  • Write the ratio of 14 and 18 in its lowest terms
    11·2 answers
  • Find value for x please and thank you​
    10·1 answer
  • HEllllppp mEeEe oNnnn thiSss plEAsssee :')
    5·2 answers
  • (2x-4)+(5x-13)+(x+4)
    5·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!