How to find relationship between expressions.

ball

New member
Joined
Jan 1, 2018
Messages
3
I have two expressions that are related. They are related by the function f.

The expressions are

  • x+sin(x)
  • x-sin(x)

They are related like so
x+sin(x) = f( x-sin(x) )


I am interested in finding the fuction f. What is the definition of f and how can that be
determined?
 
I have two expressions that are related. They are related by the function f.

The expressions are

  • x+sin(x)
  • x-sin(x)

They are related like so
x+sin(x) = f( x-sin(x) )


I am interested in finding the function f. What is the definition of f and how can that be
determined?

If we could find the inverse of the function g(x) = x - sin(x), then we could say that

f(g(x)) = x + sin(x)

and replace x with g^-1(x):

f(g(g^-1(x))) = g^-1(x) + sin(g^-1(x))

Since the left side is f(x), we would have the answer.

But I am almost certain that we can't find an inverse for g algebraically, because x appears both inside and outside the transcendental function sin. I don't think there is any way to succeed; I tried a different approach and it led to the same conclusion.

Can you tell us the source of the problem, and anything else about the context that might be useful? If it is an exercise from some textbook, knowing what they have taught would help to determine whether we can expect to solve it, and what tools might be needed. If you just made up the problem (or need it as part of something bigger) and have no specific reason to believe it is solvable, then it probably isn't. One of the "dirty little secrets" of math teaching is that most equations you can write can't be solved exactly; we only teach you how to solve the problems that can be solved, which makes algebra look more powerful than it is. Often numerical methods are needed instead (as in a graphing calculator).
 
The context

If we could find the inverse of the function g(x) = x - sin(x), then we could say that

f(g(x)) = x + sin(x)

and replace x with g^-1(x):

f(g(g^-1(x))) = g^-1(x) + sin(g^-1(x))

Since the left side is f(x), we would have the answer.

But I am almost certain that we can't find an inverse for g algebraically, because x appears both inside and outside the transcendental function sin. I don't think there is any way to succeed; I tried a different approach and it led to the same conclusion.

Can you tell us the source of the problem, and anything else about the context that might be useful? If it is an exercise from some textbook, knowing what they have taught would help to determine whether we can expect to solve it, and what tools might be needed. If you just made up the problem (or need it as part of something bigger) and have no specific reason to believe it is solvable, then it probably isn't. One of the "dirty little secrets" of math teaching is that most equations you can write can't be solved exactly; we only teach you how to solve the problems that can be solved, which makes algebra look more powerful than it is. Often numerical methods are needed instead (as in a graphing calculator).

Firstly, I want to thank you for responding so appropriately.
Secondly this is a problem of my own creation. I do, however, have reason to believe it is not impossible.

You see this problem came from my own exploration. I have been observing the effects of functions in the form of T(x,y)=(x,y). That is to say having two inputs and two outputs. My goal is to have some function f(x) and some funxtion T(x,y) and somehow find p, the result of appling T(x,y) to every point defined by f(x). I call it p and not p(x) because the curve is usually not defined by a function.

The function T(x,y)=(x-y,x+y) results in a 45 degree rotation CCW, and keeps the origin in place (if applied to all points on the plane). The function f(x)=sin(x) is well ... you know. We know y=f(x) so T(x,y)=(x-f(x),x+f(x)) , expanding f(x) will result in T(x,y)=(x-sin(x),x+sin(x)). This is how I got the expressions earlier. By reason alone, it can be determined that the graph of the function which relates the expressions is a sine wave rotated CCW by 45 degrees. Thanks to this video https://youtu.be/BPgq2AudoEo that can be found easily. Such a curve can be defined as

y=(x*cos(a)+ y*sin(a))*sin(a)+sin(x*cos(a)+y*sin(a))*cos(a)

where a is the angle to rotate by, in this case 45 degrees, or parametricly

x=t*cos(a)-sin(t)*sin(a)
y=t*sin(a)+sin(t)*cos(a)

Although it is not a true function, because neither variable can be made independent, it defines the expected curve. This, at least I think, means that from the original post, there should be a way to reach the expected curve. I have no clue how you would reach the expected curve, I've tried, but I think there should be some way.
Reading this makes me feel as if I've posted this to the wrong category, but the original post fell into this category so I think it's fine.
 
Solution

Firstly, I want to thank you for responding so appropriately.
Secondly this is a problem of my own creation. I do, however, have reason to believe it is not impossible.

You see this problem came from my own exploration. I have been observing the effects of functions in the form of T(x,y)=(x,y). That is to say having two inputs and two outputs. My goal is to have some function f(x) and some funxtion T(x,y) and somehow find p, the result of appling T(x,y) to every point defined by f(x). I call it p and not p(x) because the curve is usually not defined by a function.

The function T(x,y)=(x-y,x+y) results in a 45 degree rotation CCW, and keeps the origin in place (if applied to all points on the plane). The function f(x)=sin(x) is well ... you know. We know y=f(x) so T(x,y)=(x-f(x),x+f(x)) , expanding f(x) will result in T(x,y)=(x-sin(x),x+sin(x)). This is how I got the expressions earlier. By reason alone, it can be determined that the graph of the function which relates the expressions is a sine wave rotated CCW by 45 degrees. Thanks to this video https://youtu.be/BPgq2AudoEo that can be found easily. Such a curve can be defined as

y=(x*cos(a)+ y*sin(a))*sin(a)+sin(x*cos(a)+y*sin(a))*cos(a)

where a is the angle to rotate by, in this case 45 degrees, or parametricly

x=t*cos(a)-sin(t)*sin(a)
y=t*sin(a)+sin(t)*cos(a)

Although it is not a true function, because neither variable can be made independent, it defines the expected curve. This, at least I think, means that from the original post, there should be a way to reach the expected curve. I have no clue how you would reach the expected curve, I've tried, but I think there should be some way.
Reading this makes me feel as if I've posted this to the wrong category, but the original post fell into this category so I think it's fine.


I've got it. I was looking at this the wrong way.

To get the expected curve is very simple. The expressions define it themselves. In the function T(x,y)=(x-sin(x),x+sin(x)) the x and y are defined, all that is needed it to plot it parametricly, so

x=t-sin(t)
y=t+sin(t)

This makes a curve one would expect. One problem, its not the same curve as gotten from the other method. Upon futher inspection of the function T(x,y)=(x-y,x+y) , I've determined that in addition to a 45 degree rotation, centered at the origin, the function also scales up the graph by a factor of 2. Knowing that explains why the result is deffrent from what I originally got. Everything makes sense, the universe is explained, thank you.
 
Let's see if I can put all this together to summarize.

The expressions are
  • x+sin(x)
  • x-sin(x)

They are related like so
x+sin(x) = f( x-sin(x) )

I am interested in finding the function f. What is the definition of f and how can that be
determined?

From what you said later, it sounds like you didn't really need to find the function f; you are okay with a parametric form for the curve you are really interested in.

Secondly this is a problem of my own creation. I do, however, have reason to believe it is not impossible.

Generally, a problem you create yourself is likely to be impossible!

You see this problem came from my own exploration. I have been observing the effects of functions in the form of T(x,y)=(x,y). That is to say having two inputs and two outputs. My goal is to have some function f(x) and some function T(x,y) and somehow find p, the result of appling T(x,y) to every point defined by f(x). I call it p and not p(x) because the curve is usually not defined by a function.

What you are talking about is a transformation of the plane, which you want to apply to a curve defined by a given function f, namely y = f(x). You are right that the resulting curve will not in general be expressed as a function of x, but will have to be expressed parametrically. (Your terminology is a little off, though. f(x) is just a function; it does not define a point. What you mean to say is that the equation y=f(x) defines a curve, and that some relation p will define a curve, which will not have the form y=p(x). Be careful not to confuse these concepts.)

But in fact, it happens that rotating the sine curve by 45 degrees results in a curve that is monotonically increasing, so in principle y IS a function of x (that is, there is only one y for a given x); but that is not the same as being able to solve for y algebraically, which I think is what you want, and what I said was impossible.

The function T(x,y)=(x-y,x+y) results in a 45 degree rotation CCW, and keeps the origin in place (if applied to all points on the plane). The function f(x)=sin(x) is well ... you know.

As you mention later, it is actually a rotation together with a dilation by a factor of \(\displaystyle \sqrt{2}\).

We know y=f(x) so T(x,y)=(x-f(x),x+f(x)) , expanding f(x) will result in T(x,y)=(x-sin(x),x+sin(x)). This is how I got the expressions earlier. By reason alone, it can be determined that the graph of the function which relates the expressions is a sine wave rotated CCW by 45 degrees.

Right, apart from the dilation. You are applying the transformation T to the curve y = sin(x), that is, to the point (t, sin(t)), namely T(t, sin(t)) = (t-sin(t), t+sin(t)). This yields a point expressed in terms of the parameter t that defined a point on the original curve.

Thanks to this video https://youtu.be/BPgq2AudoEo that can be found easily. Such a curve can be defined as
y=(x*cos(a)+ y*sin(a))*sin(a)+sin(x*cos(a)+y*sin(a))*cos(a)

where a is the angle to rotate by, in this case 45 degrees, or parametricly

x=t*cos(a)-sin(t)*sin(a)
y=t*sin(a)+sin(t)*cos(a)

Although it is not a true function, because neither variable can be made independent, it defines the expected curve. This, at least I think, means that from the original post, there should be a way to reach the expected curve. I have no clue how you would reach the expected curve, I've tried, but I think there should be some way.

I didn't bother to watch the whole video, or to verify the details of what you wrote here. It should presumably be duplicating what you've done, more or less. Using a 45 degree angle, so that sin(a) = cos(a) = sqrt(2)/2, you have
x = sqrt(2)/2 * (t - sin(t))
y = sqrt(2)/2 * (t + sin(t))

as above.

But I think you are misunderstanding what a "true function" is. I think you are saying that y can't be expressed as a function of x. As I said above, that is not really true here; there is one value of y for any value of x, which is what a function is. What you MEAN seems to be that y can't be solved exactly as a function of x, which is essentially what I said in the first place. You are confusing the existence of a function with the possibility of finding it in closed form.

I've got it. I was looking at this the wrong way.

To get the expected curve is very simple. The expressions define it themselves. In the function T(x,y)=(x-sin(x),x+sin(x)) the x and y are defined, all that is needed it to plot it parametricly, so

x=t-sin(t)
y=t+sin(t)

This makes a curve one would expect. One problem, its not the same curve as gotten from the other method. Upon futher inspection of the function T(x,y)=(x-y,x+y) , I've determined that in addition to a 45 degree rotation, centered at the origin, the function also scales up the graph by a factor of 2. Knowing that explains why the result is deffrent from what I originally got. Everything makes sense, the universe is explained, thank you.

As I suggested earlier, it's best to use that parameter t from the start, and not express T(x,y) in terms of x (which is a variable in the original curve's definition, not the new one).

Yes, I think everything is cleared up.
 
Last edited:
Top