How to solve functions by having in-function equations (bad explanation, but read body)

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Thinking about this all day, still don't know how to solve it:
Find $f(x)$ if
$$fleft(x+1 over x-2right)-2fleft(x-2 over x+1right)=x$$
........

algebra-precalculus functions

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edited 1 hour ago

Avinash N

1909

asked 1 hour ago

Aleksa

182

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up vote
2
down vote

favorite

Thinking about this all day, still don't know how to solve it:
Find $f(x)$ if
$$fleft(x+1 over x-2right)-2fleft(x-2 over x+1right)=x$$
........

algebra-precalculus functions

share|cite|improve this question

edited 1 hour ago

Avinash N

1909

asked 1 hour ago

Aleksa

182

New contributor

Aleksa is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

add a comment | 

up vote
2
down vote

favorite

up vote
2
down vote

favorite

Thinking about this all day, still don't know how to solve it:
Find $f(x)$ if
$$fleft(x+1 over x-2right)-2fleft(x-2 over x+1right)=x$$
........

algebra-precalculus functions

share|cite|improve this question

edited 1 hour ago

Avinash N

1909

asked 1 hour ago

Aleksa

182

New contributor

Aleksa is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

Thinking about this all day, still don't know how to solve it:
Find $f(x)$ if
$$fleft(x+1 over x-2right)-2fleft(x-2 over x+1right)=x$$
........

algebra-precalculus functions

algebra-precalculus functions

share|cite|improve this question

edited 1 hour ago

Avinash N

1909

asked 1 hour ago

Aleksa

182

New contributor

Aleksa is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

share|cite|improve this question

edited 1 hour ago

Avinash N

1909

asked 1 hour ago

Aleksa

182

New contributor

Aleksa is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

share|cite|improve this question

share|cite|improve this question

edited 1 hour ago

Avinash N

1909

edited 1 hour ago

Avinash N

1909

edited 1 hour ago

Avinash N

1909

1909

asked 1 hour ago

Aleksa

182

New contributor

Aleksa is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked 1 hour ago

Aleksa

182

asked 1 hour ago

Aleksa

182

182

New contributor

Aleksa is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

New contributor

Aleksa is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

Aleksa is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

add a comment | 

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4 Answers
4

active

oldest

votes

up vote
5
down vote



accepted

Write $t = (x+1)/(x-2)$ then $x= (2t+1)/(t-1)$ and $$f(t)+2f(1over t) = 2t+1over t-1$$

Replaceing $t$ with $1/t$ we get $$f(1over t)+2f(t) = 2+tover 1-t$$

Solving this system on $f(t)$ we get:

$$ 22+tover 1-t-2t+1over t-1 =3f(t)$$

and we are done...

share|cite|improve this answer

edited 1 hour ago

answered 1 hour ago

greedoid

30.4k94183

• 
  
  
  

  
  

  
  
  
  
  
  

  
  I don't get what you did at the second step, the replacing of t and t/1, could you explain it?
  – Aleksa
  1 hour ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Everywhere in first equation replace t with 1/t
  – greedoid
  1 hour ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I'm not sure what you don't understand.
  – greedoid
  1 hour ago
  

  
  
  
  

add a comment | 

up vote
2
down vote

Hint

Solve:

$$
f(y) +2fleft(frac 1yright) = phi(y)\
fleft(frac 1yright)+2f(y) = psi(y)
$$

with $y = fracx+1x-2$

share|cite|improve this answer

edited 1 hour ago

answered 1 hour ago

Cesareo

6,4292413

add a comment | 

up vote
1
down vote

If
$$fleft(x+1 over x-2right)-2fleft(x-2 over x+1right)=x$$

Then I got,

$f(x)=$$1over x-1$

share|cite|improve this answer

answered 1 hour ago

Avinash N

1909

add a comment | 

up vote
0
down vote

By making the transformation $x to 1-x$ this may be seen as follows:

For
$$fleft(x+1 over x-2right)-2fleft(x-2 over x+1right)=x$$
let $x to 1-x$ to obtain
$$fleft(x-2 over x+1right)-2fleft(x+1 over x-2right)=1-x,$$
or
$$fleft(fracx-2x+1right) = 2 , fleft(fracx+1x-2right) + 1-x.$$
From this it is seen that
$$fleft(fracx+1x-2right) = fracx-23 = frac1left(fracx+1x-2right) - 1.$$
This leads to
$$f(t) = frac1t-1.$$

share|cite|improve this answer

answered 23 mins ago

Leucippus

19.2k102869

add a comment | 

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4 Answers
4

active

oldest

votes

4 Answers
4

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
5
down vote



accepted

Write $t = (x+1)/(x-2)$ then $x= (2t+1)/(t-1)$ and $$f(t)+2f(1over t) = 2t+1over t-1$$

Replaceing $t$ with $1/t$ we get $$f(1over t)+2f(t) = 2+tover 1-t$$

Solving this system on $f(t)$ we get:

$$ 22+tover 1-t-2t+1over t-1 =3f(t)$$

and we are done...

share|cite|improve this answer

edited 1 hour ago

answered 1 hour ago

greedoid

30.4k94183

• 
  
  
  

  
  

  
  
  
  
  
  

  
  I don't get what you did at the second step, the replacing of t and t/1, could you explain it?
  – Aleksa
  1 hour ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Everywhere in first equation replace t with 1/t
  – greedoid
  1 hour ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I'm not sure what you don't understand.
  – greedoid
  1 hour ago
  

  
  
  
  

add a comment | 

up vote
5
down vote



accepted

Write $t = (x+1)/(x-2)$ then $x= (2t+1)/(t-1)$ and $$f(t)+2f(1over t) = 2t+1over t-1$$

Replaceing $t$ with $1/t$ we get $$f(1over t)+2f(t) = 2+tover 1-t$$

Solving this system on $f(t)$ we get:

$$ 22+tover 1-t-2t+1over t-1 =3f(t)$$

and we are done...

share|cite|improve this answer

edited 1 hour ago

answered 1 hour ago

greedoid

30.4k94183

• 
  
  
  

  
  

  
  
  
  
  
  

  
  I don't get what you did at the second step, the replacing of t and t/1, could you explain it?
  – Aleksa
  1 hour ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Everywhere in first equation replace t with 1/t
  – greedoid
  1 hour ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I'm not sure what you don't understand.
  – greedoid
  1 hour ago
  

  
  
  
  

add a comment | 

up vote
5
down vote



accepted

up vote
5
down vote



accepted

Write $t = (x+1)/(x-2)$ then $x= (2t+1)/(t-1)$ and $$f(t)+2f(1over t) = 2t+1over t-1$$

Replaceing $t$ with $1/t$ we get $$f(1over t)+2f(t) = 2+tover 1-t$$

Solving this system on $f(t)$ we get:

$$ 22+tover 1-t-2t+1over t-1 =3f(t)$$

and we are done...

share|cite|improve this answer

edited 1 hour ago

answered 1 hour ago

greedoid

30.4k94183

Write $t = (x+1)/(x-2)$ then $x= (2t+1)/(t-1)$ and $$f(t)+2f(1over t) = 2t+1over t-1$$

Replaceing $t$ with $1/t$ we get $$f(1over t)+2f(t) = 2+tover 1-t$$

Solving this system on $f(t)$ we get:

$$ 22+tover 1-t-2t+1over t-1 =3f(t)$$

and we are done...

share|cite|improve this answer

edited 1 hour ago

answered 1 hour ago

greedoid

30.4k94183

share|cite|improve this answer

share|cite|improve this answer

edited 1 hour ago

edited 1 hour ago

edited 1 hour ago

answered 1 hour ago

greedoid

30.4k94183

answered 1 hour ago

greedoid

30.4k94183

answered 1 hour ago

greedoid

30.4k94183

30.4k94183

• 
  
  
  

  
  

  
  
  
  
  
  

  
  I don't get what you did at the second step, the replacing of t and t/1, could you explain it?
  – Aleksa
  1 hour ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Everywhere in first equation replace t with 1/t
  – greedoid
  1 hour ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I'm not sure what you don't understand.
  – greedoid
  1 hour ago
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  I don't get what you did at the second step, the replacing of t and t/1, could you explain it?
  – Aleksa
  1 hour ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Everywhere in first equation replace t with 1/t
  – greedoid
  1 hour ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I'm not sure what you don't understand.
  – greedoid
  1 hour ago
  

  
  
  
  

I don't get what you did at the second step, the replacing of t and t/1, could you explain it?
– Aleksa
1 hour ago

I don't get what you did at the second step, the replacing of t and t/1, could you explain it?
– Aleksa
1 hour ago

Everywhere in first equation replace t with 1/t
– greedoid
1 hour ago

Everywhere in first equation replace t with 1/t
– greedoid
1 hour ago

I'm not sure what you don't understand.
– greedoid
1 hour ago

I'm not sure what you don't understand.
– greedoid
1 hour ago

add a comment | 

up vote
2
down vote

Hint

Solve:

$$
f(y) +2fleft(frac 1yright) = phi(y)\
fleft(frac 1yright)+2f(y) = psi(y)
$$

with $y = fracx+1x-2$

share|cite|improve this answer

edited 1 hour ago

answered 1 hour ago

Cesareo

6,4292413

add a comment | 

up vote
2
down vote

Hint

Solve:

$$
f(y) +2fleft(frac 1yright) = phi(y)\
fleft(frac 1yright)+2f(y) = psi(y)
$$

with $y = fracx+1x-2$

share|cite|improve this answer

edited 1 hour ago

answered 1 hour ago

Cesareo

6,4292413

add a comment | 

up vote
2
down vote

up vote
2
down vote

Hint

Solve:

$$
f(y) +2fleft(frac 1yright) = phi(y)\
fleft(frac 1yright)+2f(y) = psi(y)
$$

with $y = fracx+1x-2$

share|cite|improve this answer

edited 1 hour ago

answered 1 hour ago

Cesareo

6,4292413

Hint

Solve:

$$
f(y) +2fleft(frac 1yright) = phi(y)\
fleft(frac 1yright)+2f(y) = psi(y)
$$

with $y = fracx+1x-2$

share|cite|improve this answer

edited 1 hour ago

answered 1 hour ago

Cesareo

6,4292413

share|cite|improve this answer

share|cite|improve this answer

edited 1 hour ago

edited 1 hour ago

edited 1 hour ago

answered 1 hour ago

Cesareo

6,4292413

answered 1 hour ago

Cesareo

6,4292413

answered 1 hour ago

Cesareo

6,4292413

6,4292413

add a comment | 

add a comment | 

up vote
1
down vote

If
$$fleft(x+1 over x-2right)-2fleft(x-2 over x+1right)=x$$

Then I got,

$f(x)=$$1over x-1$

share|cite|improve this answer

answered 1 hour ago

Avinash N

1909

add a comment | 

up vote
1
down vote

If
$$fleft(x+1 over x-2right)-2fleft(x-2 over x+1right)=x$$

Then I got,

$f(x)=$$1over x-1$

share|cite|improve this answer

answered 1 hour ago

Avinash N

1909

add a comment | 

up vote
1
down vote

up vote
1
down vote

If
$$fleft(x+1 over x-2right)-2fleft(x-2 over x+1right)=x$$

Then I got,

$f(x)=$$1over x-1$

share|cite|improve this answer

answered 1 hour ago

Avinash N

1909

If
$$fleft(x+1 over x-2right)-2fleft(x-2 over x+1right)=x$$

Then I got,

$f(x)=$$1over x-1$

share|cite|improve this answer

answered 1 hour ago

Avinash N

1909

share|cite|improve this answer

share|cite|improve this answer

answered 1 hour ago

Avinash N

1909

answered 1 hour ago

Avinash N

1909

answered 1 hour ago

Avinash N

1909

1909

add a comment | 

add a comment | 

up vote
0
down vote

By making the transformation $x to 1-x$ this may be seen as follows:

For
$$fleft(x+1 over x-2right)-2fleft(x-2 over x+1right)=x$$
let $x to 1-x$ to obtain
$$fleft(x-2 over x+1right)-2fleft(x+1 over x-2right)=1-x,$$
or
$$fleft(fracx-2x+1right) = 2 , fleft(fracx+1x-2right) + 1-x.$$
From this it is seen that
$$fleft(fracx+1x-2right) = fracx-23 = frac1left(fracx+1x-2right) - 1.$$
This leads to
$$f(t) = frac1t-1.$$

share|cite|improve this answer

answered 23 mins ago

Leucippus

19.2k102869

add a comment | 

up vote
0
down vote

By making the transformation $x to 1-x$ this may be seen as follows:

For
$$fleft(x+1 over x-2right)-2fleft(x-2 over x+1right)=x$$
let $x to 1-x$ to obtain
$$fleft(x-2 over x+1right)-2fleft(x+1 over x-2right)=1-x,$$
or
$$fleft(fracx-2x+1right) = 2 , fleft(fracx+1x-2right) + 1-x.$$
From this it is seen that
$$fleft(fracx+1x-2right) = fracx-23 = frac1left(fracx+1x-2right) - 1.$$
This leads to
$$f(t) = frac1t-1.$$

share|cite|improve this answer

answered 23 mins ago

Leucippus

19.2k102869

add a comment | 

up vote
0
down vote

up vote
0
down vote

By making the transformation $x to 1-x$ this may be seen as follows:

For
$$fleft(x+1 over x-2right)-2fleft(x-2 over x+1right)=x$$
let $x to 1-x$ to obtain
$$fleft(x-2 over x+1right)-2fleft(x+1 over x-2right)=1-x,$$
or
$$fleft(fracx-2x+1right) = 2 , fleft(fracx+1x-2right) + 1-x.$$
From this it is seen that
$$fleft(fracx+1x-2right) = fracx-23 = frac1left(fracx+1x-2right) - 1.$$
This leads to
$$f(t) = frac1t-1.$$

share|cite|improve this answer

answered 23 mins ago

Leucippus

19.2k102869

By making the transformation $x to 1-x$ this may be seen as follows:

For
$$fleft(x+1 over x-2right)-2fleft(x-2 over x+1right)=x$$
let $x to 1-x$ to obtain
$$fleft(x-2 over x+1right)-2fleft(x+1 over x-2right)=1-x,$$
or
$$fleft(fracx-2x+1right) = 2 , fleft(fracx+1x-2right) + 1-x.$$
From this it is seen that
$$fleft(fracx+1x-2right) = fracx-23 = frac1left(fracx+1x-2right) - 1.$$
This leads to
$$f(t) = frac1t-1.$$

share|cite|improve this answer

answered 23 mins ago

Leucippus

19.2k102869

share|cite|improve this answer

share|cite|improve this answer

answered 23 mins ago

Leucippus

19.2k102869

answered 23 mins ago

Leucippus

19.2k102869

answered 23 mins ago

Leucippus

19.2k102869

19.2k102869

add a comment | 

add a comment | 

Aleksa is a new contributor. Be nice, and check out our Code of Conduct.

 

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