Splitting function into odd and even parts

  • #1
j3dwards
32
0

Homework Statement


Split the function f(x) = ex + 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:

ex = 0.5[ex - e-x] + 0.5[ex + 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?
 

Answers and Replies

  • #2
Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
10,706
1,722

Homework Statement


Split the function f(x) = ex + 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:

ex = 0.5[ex - e-x] + 0.5[ex + 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?
 
  • #3
j3dwards
32
0
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.
 
  • #4
LCKurtz
Science Advisor
Homework Helper
Insights Author
Gold Member
9,568
774
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.
 
  • #5
j3dwards
32
0
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?
 
  • #6
Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
10,706
1,722
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.
 
  • #7
Thewindyfan
68
19
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.
 
  • #8
j3dwards
32
0
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.
 
  • #9
j3dwards
32
0
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?
 
  • #10
Thewindyfan
68
19
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
 
  • #11
j3dwards
32
0
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.
 

Suggested for: Splitting function into odd and even parts

  • Last Post
Replies
5
Views
1K
Replies
6
Views
14K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
18
Views
13K
  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
2
Views
7K
Replies
1
Views
1K
  • Last Post
Replies
4
Views
1K
Replies
0
Views
1K
Replies
4
Views
4K
Top