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Find $f(5)$ where $f$ satisfies $f(x)+f(1/(1-x))=x $

A function satisfies the identity $f(x) + 2fleft(frac1xright) = 2x+1$ … find another identity that $f(x)$ satisfies.The function $f (x) = f left (frac x2 right ) + f left (frac x2 + frac 12right)$A function $f$ satisfies the condition $f[f(x) - e^x] = e + 1$ for all $x in Bbb R$.Changing $y=mx+b$ equation into $ax+by=c$Finding the sums of all the solution for values of $z$ where $f(3z)=7$ in $f(fracx3)=x^2+x+1$Find the maximum area using an unspecified pronumeral value onlyRearranging formula to find desired variableIf $f(x)$ satisfies $2f (x) = f(xy) + f(x/y)$, find $f(x)$Amplitude of mass-spring system after application of impulse forcing functionHow to solve equations in the form $af^2(x)+bf(x)+cx=0$?

4

2

$begingroup$

Question:

How do you Find $f(5)$ in which the function satisfies
$$f(x)+fleft(frac11-xright)=x $$
where $xinBbbR$ and $xneq 0,1$?

---

My steps:

Step 1)

Substitute $5$ into the equation to get:

$$f(5)+fleft(frac1-4right)=5$$

But then I had gotten stuck there and I could not find $f(5)$

Please write detailed steps.

algebra-precalculus functional-equations

share|cite|improve this question

edited Mar 28 at 17:25

Jyrki Lahtonen

110k13172389

asked Jul 6 '16 at 7:17

user304703

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Please write your question properly !!!
  $endgroup$
  – Vineet Mangal
  Jul 6 '16 at 7:19
  

  
  
  
  

add a comment | 

4

2

$begingroup$

Question:

How do you Find $f(5)$ in which the function satisfies
$$f(x)+fleft(frac11-xright)=x $$
where $xinBbbR$ and $xneq 0,1$?

---

My steps:

Step 1)

Substitute $5$ into the equation to get:

$$f(5)+fleft(frac1-4right)=5$$

But then I had gotten stuck there and I could not find $f(5)$

Please write detailed steps.

algebra-precalculus functional-equations

share|cite|improve this question

edited Mar 28 at 17:25

Jyrki Lahtonen

110k13172389

asked Jul 6 '16 at 7:17

user304703

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Please write your question properly !!!
  $endgroup$
  – Vineet Mangal
  Jul 6 '16 at 7:19
  

  
  
  
  

add a comment | 

$begingroup$

Question:

How do you Find $f(5)$ in which the function satisfies
$$f(x)+fleft(frac11-xright)=x $$
where $xinBbbR$ and $xneq 0,1$?

---

My steps:

Step 1)

Substitute $5$ into the equation to get:

$$f(5)+fleft(frac1-4right)=5$$

But then I had gotten stuck there and I could not find $f(5)$

Please write detailed steps.

algebra-precalculus functional-equations

share|cite|improve this question

edited Mar 28 at 17:25

Jyrki Lahtonen

110k13172389

asked Jul 6 '16 at 7:17

user304703

$endgroup$

share|cite|improve this question

edited Mar 28 at 17:25

Jyrki Lahtonen

110k13172389

asked Jul 6 '16 at 7:17

user304703

share|cite|improve this question

edited Mar 28 at 17:25

Jyrki Lahtonen

110k13172389

asked Jul 6 '16 at 7:17

user304703

edited Mar 28 at 17:25

Jyrki Lahtonen

110k13172389

edited Mar 28 at 17:25

Jyrki Lahtonen

110k13172389

3 Answers
3

active

oldest

votes

1

$begingroup$

by using $x=5,4/5,-1/4$ we have:
$$fleft(-frac 1 4right)+fleft(frac 4 5right)=-frac 1 4$$
$$ f(5)+fleft(frac 4 5right)=frac 4 5$$
$$f(5)+fleft(-frac 1 4right)=5$$

Then sum the last two expressions and subtract the first to get:
$$ 2f(5)+fleft(frac 4 5right)+fleft(-frac 1 4right)-fleft(-frac 1 4right)-fleft(frac 4 5right)=5+frac 45+frac 1 4$$
hence $2f(5)=6.05$ and then $f(5)=3.025$.

share|cite|improve this answer

edited Jul 6 '16 at 20:00

Solomonoff's Secret

3,65211233

answered Jul 6 '16 at 7:28

SpottySpotty

80959

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  It's worth noting that generalising this gives you an way of finding that $f(x)=0.5(x-1/(1-x)+1-1/x)$.
  $endgroup$
  – YiFan
  Mar 28 at 23:28
  

  
  
  
  

add a comment | 

11

$begingroup$

$$fleft(5right)+fleft(-frac14right)=5$$

If we know $fleft(-frac14right)$, we can solve the problem.

$$fleft(-frac14right)+fleft(frac45right)=-frac14$$

If we know $fleft(frac45 right)$, we can solve the problem.

$$fleft(frac45right)+fleft(5right)=frac45$$

Why don't we just solve the linear system? Are you able to solve it?

share|cite|improve this answer

answered Jul 6 '16 at 7:28

Siong Thye GohSiong Thye Goh

103k1468120

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I am confused because when you substitute (-1/4) into f(1/(1−x)) you should get (5/4)?
  $endgroup$
  – user304703
  Jul 6 '16 at 7:41
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  $$frac11-left(-frac14right)=frac11+left(frac14right)=4/5$$
  $endgroup$
  – Siong Thye Goh
  Jul 6 '16 at 7:44
  
  

  
  
  
  

add a comment | 

2

$begingroup$

You can use the fact that$$left( frac11-x right)^-1=1-frac1x,$$where the exponent $-1$ stands for the reverse. If you substitute $1/(1-x)$ and $1-1/x$ in the functional equation and solve three simultaneous equations, you can find general form of $f(x)$.

share|cite|improve this answer

answered Jul 6 '16 at 7:34

GhartalGhartal

2,87911435

$endgroup$

add a comment | 

Your Answer

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1

$begingroup$

by using $x=5,4/5,-1/4$ we have:
$$fleft(-frac 1 4right)+fleft(frac 4 5right)=-frac 1 4$$
$$ f(5)+fleft(frac 4 5right)=frac 4 5$$
$$f(5)+fleft(-frac 1 4right)=5$$

Then sum the last two expressions and subtract the first to get:
$$ 2f(5)+fleft(frac 4 5right)+fleft(-frac 1 4right)-fleft(-frac 1 4right)-fleft(frac 4 5right)=5+frac 45+frac 1 4$$
hence $2f(5)=6.05$ and then $f(5)=3.025$.

share|cite|improve this answer

edited Jul 6 '16 at 20:00

Solomonoff's Secret

3,65211233

answered Jul 6 '16 at 7:28

SpottySpotty

80959

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  It's worth noting that generalising this gives you an way of finding that $f(x)=0.5(x-1/(1-x)+1-1/x)$.
  $endgroup$
  – YiFan
  Mar 28 at 23:28
  

  
  
  
  

add a comment | 

1

$begingroup$

by using $x=5,4/5,-1/4$ we have:
$$fleft(-frac 1 4right)+fleft(frac 4 5right)=-frac 1 4$$
$$ f(5)+fleft(frac 4 5right)=frac 4 5$$
$$f(5)+fleft(-frac 1 4right)=5$$

Then sum the last two expressions and subtract the first to get:
$$ 2f(5)+fleft(frac 4 5right)+fleft(-frac 1 4right)-fleft(-frac 1 4right)-fleft(frac 4 5right)=5+frac 45+frac 1 4$$
hence $2f(5)=6.05$ and then $f(5)=3.025$.

share|cite|improve this answer

edited Jul 6 '16 at 20:00

Solomonoff's Secret

3,65211233

answered Jul 6 '16 at 7:28

SpottySpotty

80959

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  It's worth noting that generalising this gives you an way of finding that $f(x)=0.5(x-1/(1-x)+1-1/x)$.
  $endgroup$
  – YiFan
  Mar 28 at 23:28
  

  
  
  
  

add a comment | 

$begingroup$

by using $x=5,4/5,-1/4$ we have:
$$fleft(-frac 1 4right)+fleft(frac 4 5right)=-frac 1 4$$
$$ f(5)+fleft(frac 4 5right)=frac 4 5$$
$$f(5)+fleft(-frac 1 4right)=5$$

Then sum the last two expressions and subtract the first to get:
$$ 2f(5)+fleft(frac 4 5right)+fleft(-frac 1 4right)-fleft(-frac 1 4right)-fleft(frac 4 5right)=5+frac 45+frac 1 4$$
hence $2f(5)=6.05$ and then $f(5)=3.025$.

share|cite|improve this answer

edited Jul 6 '16 at 20:00

Solomonoff's Secret

3,65211233

answered Jul 6 '16 at 7:28

SpottySpotty

80959

$endgroup$

share|cite|improve this answer

edited Jul 6 '16 at 20:00

Solomonoff's Secret

3,65211233

answered Jul 6 '16 at 7:28

SpottySpotty

80959

edited Jul 6 '16 at 20:00

Solomonoff's Secret

3,65211233

edited Jul 6 '16 at 20:00

Solomonoff's Secret

3,65211233

answered Jul 6 '16 at 7:28

SpottySpotty

80959

answered Jul 6 '16 at 7:28

SpottySpotty

80959

11

$begingroup$

$$fleft(5right)+fleft(-frac14right)=5$$

If we know $fleft(-frac14right)$, we can solve the problem.

$$fleft(-frac14right)+fleft(frac45right)=-frac14$$

If we know $fleft(frac45 right)$, we can solve the problem.

$$fleft(frac45right)+fleft(5right)=frac45$$

Why don't we just solve the linear system? Are you able to solve it?

share|cite|improve this answer

answered Jul 6 '16 at 7:28

Siong Thye GohSiong Thye Goh

103k1468120

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I am confused because when you substitute (-1/4) into f(1/(1−x)) you should get (5/4)?
  $endgroup$
  – user304703
  Jul 6 '16 at 7:41
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  $$frac11-left(-frac14right)=frac11+left(frac14right)=4/5$$
  $endgroup$
  – Siong Thye Goh
  Jul 6 '16 at 7:44
  
  

  
  
  
  

add a comment | 

11

$begingroup$

$$fleft(5right)+fleft(-frac14right)=5$$

If we know $fleft(-frac14right)$, we can solve the problem.

$$fleft(-frac14right)+fleft(frac45right)=-frac14$$

If we know $fleft(frac45 right)$, we can solve the problem.

$$fleft(frac45right)+fleft(5right)=frac45$$

Why don't we just solve the linear system? Are you able to solve it?

share|cite|improve this answer

answered Jul 6 '16 at 7:28

Siong Thye GohSiong Thye Goh

103k1468120

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I am confused because when you substitute (-1/4) into f(1/(1−x)) you should get (5/4)?
  $endgroup$
  – user304703
  Jul 6 '16 at 7:41
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  $$frac11-left(-frac14right)=frac11+left(frac14right)=4/5$$
  $endgroup$
  – Siong Thye Goh
  Jul 6 '16 at 7:44
  
  

  
  
  
  

add a comment | 

$begingroup$

$$fleft(5right)+fleft(-frac14right)=5$$

If we know $fleft(-frac14right)$, we can solve the problem.

$$fleft(-frac14right)+fleft(frac45right)=-frac14$$

If we know $fleft(frac45 right)$, we can solve the problem.

$$fleft(frac45right)+fleft(5right)=frac45$$

Why don't we just solve the linear system? Are you able to solve it?

share|cite|improve this answer

answered Jul 6 '16 at 7:28

Siong Thye GohSiong Thye Goh

103k1468120

$endgroup$

share|cite|improve this answer

answered Jul 6 '16 at 7:28

Siong Thye GohSiong Thye Goh

103k1468120

answered Jul 6 '16 at 7:28

Siong Thye GohSiong Thye Goh

103k1468120

answered Jul 6 '16 at 7:28

Siong Thye GohSiong Thye Goh

103k1468120

2

$begingroup$

You can use the fact that$$left( frac11-x right)^-1=1-frac1x,$$where the exponent $-1$ stands for the reverse. If you substitute $1/(1-x)$ and $1-1/x$ in the functional equation and solve three simultaneous equations, you can find general form of $f(x)$.

share|cite|improve this answer

answered Jul 6 '16 at 7:34

GhartalGhartal

2,87911435

$endgroup$

add a comment | 

2

$begingroup$

You can use the fact that$$left( frac11-x right)^-1=1-frac1x,$$where the exponent $-1$ stands for the reverse. If you substitute $1/(1-x)$ and $1-1/x$ in the functional equation and solve three simultaneous equations, you can find general form of $f(x)$.

share|cite|improve this answer

answered Jul 6 '16 at 7:34

GhartalGhartal

2,87911435

$endgroup$

add a comment | 

$begingroup$

You can use the fact that$$left( frac11-x right)^-1=1-frac1x,$$where the exponent $-1$ stands for the reverse. If you substitute $1/(1-x)$ and $1-1/x$ in the functional equation and solve three simultaneous equations, you can find general form of $f(x)$.

share|cite|improve this answer

answered Jul 6 '16 at 7:34

GhartalGhartal

2,87911435

$endgroup$

share|cite|improve this answer

answered Jul 6 '16 at 7:34

GhartalGhartal

2,87911435

answered Jul 6 '16 at 7:34

GhartalGhartal

2,87911435

answered Jul 6 '16 at 7:34

GhartalGhartal

2,87911435