Sum of even and odd function

[Vertciel]

Hello there,

I would like to ask for help with understanding my textbook's solution for the following problem. I have also shown my work for trying to understand it below.

Thank you very much!

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Show that every function defined for all real numbers can be written as the sum of an even and odd function.

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My Work:


\(\displaystyle f(-x) = \frac{1}{2}\left( {f(-x) + f( x)} \right) + \frac{1}{2}\left( {f(-x) - f(x)} \right)\)

Although the textbook states the grouped term on the left side is even and the one on the right is odd, my f(-x) above doesn't seem to produce anything useful as I do not know whether f(x) itself is even, odd, or neither.

 

[mmm4444bot]

I think that you need to take cases.

In other words, assume the following.

Case I

f is an even function.

Now work through the given equation based on this assumption, and show that the given equation simplifies to f(x) = f(x).

Case II

f is an odd function.

Do the same thing based on this assumption.

Case III

f is neither an even nor odd function.

This is more challenging to do symbolically.

Try working with some actual function definitions first, to get a handle on what's happening.

For example: f(x) = 5x - 6

 

[pka]

Show that every function defined for all real numbers can be written as the sum of an even and odd function.
\(\displaystyle f(x) = \frac{1}{2}\left( {f(x) + f( - x)} \right) + \frac{1}{2}\left( {f(x) - f( - x)} \right)\)
I do not know whether f(x) itself is even, odd, or neither.

You can never know that. Because the function itself can be even, odd, or neither.
The point of this exercise is simply to show that any function, even, odd, or neither, can be expressed as the sum of an even function and an odd function.

 

[PAULK]

Hello there,

I would like to ask for help with understanding my textbook's solution for the following problem. I have also shown my work for trying to understand it below.

Thank you very much!

---
Show that every function defined for all real numbers can be written as the sum of an even and odd function.

\(\displaystyle f(x) = \frac{1}{2}\left( {f(x) + f( - x)} \right) + \frac{1}{2}\left( {f(x) - f( - x)} \right)\)

---
My Work:


\(\displaystyle f(-x) = \frac{1}{2}\left( {f(-x) + f( x)} \right) + \frac{1}{2}\left( {f(-x) - f(x)} \right)\)

Although the textbook states the grouped term on the left side is even and the one on the right is odd, my f(-x) above doesn't seem to produce anything useful as I do not know whether f(x) itself is even, odd, or neither.

Of course you don't -- the exercise says that ANY function can be written, etc.

So let:

Even-part-of-F(x) = 1/2 (f(x) + f(-x))

Then show that:

Even-part-of-F(-x) = Even-part-of-F(x)

Likewise, let:

Odd-part-of-F(x) = 1/2 (f(x) - f(-x))

and show that

Odd-part-of-F(-x) = - Odd-part-of-F(x)

So these two are even and odd functions.

Finally, show that their sum comes out to f(x)

 

[Vertciel]

Thank you very much for all the responses. I was able to show the necessary steps.

I just have one more question. How would I get to the function as the textbook's solution states from scratch?

My textbook only asked for the statement to be proved and did not offer any hints as to what I should do.