Splitting function into odd and even parts

[j3dwards]

Homework Statement

Split the function f(x) = e^x + πe^−x into odd and even parts, and express your result in terms of cosh x and sinh x.

Homework Equations

f(x) = 0.5[f(x) + f(-x)] +0.5[f(x) - f(-x)]

The Attempt at a Solution

So i know that:

e^x = 0.5[e^x - e^-x] + 0.5[e^x + e^-x] = sinh(x) + cosh(x)

So let the even part be a(x) and the odd be b(x).

a(x) = cosh(x) + πcosh(x) = (1 + π) cosh(x)
b(x) = sinh(x) - πsinh(x) = (1 - π) sinh(x)

Is this correct?

 

[Ray Vickson]

Homework Statement

Split the function f(x) = e^x + πe^−x into odd and even parts, and express your result in terms of cosh x and sinh x.

Homework Equations

f(x) = 0.5[f(x) + f(-x)] +0.5[f(x) - f(-x)]

The Attempt at a Solution

So i know that:

e^x = 0.5[e^x - e^-x] + 0.5[e^x + e^-x] = sinh(x) + cosh(x)

So let the even part be a(x) and the odd be b(x).

a(x) = cosh(x) + πcosh(x) = (1 + π) cosh(x)
b(x) = sinh(x) - πsinh(x) = (1 - π) sinh(x)

Is this correct?

What do YOU think? Have you made any errors?

 

[j3dwards]

What do YOU think? Have you made any errors?

From my workings, I believe I am correct? I can't see any errors myself, so was wondering if you could help me.

 

[LCKurtz]

From my workings, I believe I am correct? I can't see any errors myself, so was wondering if you could help me.

All you have to do to check it yourself is check whether a(x) is even, b(x) is odd, and they add to your original function.

 

[j3dwards]

All you have to do to check it yourself is check whether a(x) is even, b(x) is odd, and they add to your original function.

Yes it does add up. So I am correct?

 

[Ray Vickson]

Yes it does add up. So I am correct?

Why do you need to ask? Is a(x) even? Is b(x) odd? Is it true that f(x) = a(x) + b(x)?

You should get in the habit of checking these things for yourself because in many situations (such as in exams) you cannot ask anybody else! Have some confidence in your own work.

 

[Thewindyfan]

You can double check by seeing if a(-x) = a(x) or -a(x).

Same for b(x).

Make sure to go through your steps again and see if they make sense to you.

 

[j3dwards]

Why do you need to ask? Is a(x) even? Is b(x) odd? Is it true that f(x) = a(x) + b(x)?

You should get in the habit of checking these things for yourself because in many situations (such as in exams) you cannot ask anybody else! Have some confidence in your own work.

I did check originally, but I'm just quite unsure about odd and even functions so I was just asking to make sure so that for that exam, I knew the correct method.

 

[j3dwards]

You can double check by seeing if a(-x) = a(x) or -a(x).

Same for b(x).

Make sure to go through your steps again and see if they make sense to you.

I'm not really sure how to use a(-x) = a(x) or -a(x) to check. Do I literally just make the rhs of the equation equal to negative of what it is a see if it comes out with the same answer?

 

[Thewindyfan]

I'm not really sure how to use a(-x) = a(x) or -a(x) to check.

Yep, you plug in -x and simplify the function as best as you can until you get either the original function or the opposite of the function

 

[j3dwards]

Yep, you plug in -x and simplify the function as best as you can until you get either the original function or the opposite of the function

Okay perfect, thank you so much.