Prove that (f+g)(x) is an odd function, if f and g are odd functions (Stewart, Calculus)

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Suppose f(x) and g(x) are odd functions. Prove that (f+g)(x) is also an odd function.  Answer:  1. Strategy By definition, f is an odd function if and only if f(-x) = - f(x) To show (f+g) is an odd function, we need to show (f+g)(-x) = - (f+g)(x) 2. Explanation Since $f(x)$ and $g(x)$ are odd functions $\Rightarrow f(-x) =-f(x)$ and $g(-x) =-g(x)$ By definition of sum of functions. $(f+g)(-x) =f(-x)+g(-x)$ $=-f(x)-g(x)$ $=-(f(x)+g(x))$ $=-(f+g)(x)$ (by definition of sum of functions) $\Rightarrow(f+g)(-x) =-(f+g)(x)$ $\Rightarrow f+g$ is an odd function. Q.E.D. 
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[Python] Sublime Text: set a build on Python 2.7 creating a sublime-build file (solution to WinError 2 problem)

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If anyone attempts to install Sublime Text, faces an error message [WinError 2], and notices your Python code does not work (like I did), I suggest you to do the following:

(note: I am working this on Windows 10) 

Situation: error message on my computer when I run "Build" by pressing ctrl+B 


1. Make sure your Python code is saved (e.g. I saved my HelloWorld.py file on my C:\Brian\PythonLab) 

2. Go to Tools > Build System > New Build System

3. on the untitled.sublime-build, type the following

{
    "cmd": ["(your directory for Python goes here)", "-u", "$file"],
    "file_regex": "^[ ]*File \"(...*?)\", line ([0-9]*)",
    "selector": "source.python"
}
In my case, I installed Python 2.7 on the directory C:\python27

Note that the directory should be represented using double \ (\\). 

That is,

C:\\Python27\\python.exe 

So in my case:

{
"cmd": ["C:\\Python27\\python.exe", "-u", "$file"],
"file_regex": "^[ ]*File \"(...*?)\", line ([0-9]*)",
"selector": "source.python"
}

should be the appropriate address for python.exe on my computer. 


4. save the sublime-build file: make sure the file extension must be .sublime-build

In my case, I typed "Python27.sublime-build" and saved. 

Note: make sure you save on:

C:\Users\[username]\AppData\Roaming\Sublime Text\Packages\User

 

In my case, it would be: 

C:\Users\bjhle\AppData\Roaming\Sublime Text\Packages\User

 

5. Go to Tools > Build System > Python27 (depending on the name you saved) 



6. Go to your "py" code tab > tools > build with > choose your Python (in my case Python27)


7. Run "Build," using Tools > Build or pressing Ctrl+B

 

8. Result on my Sublime Text:








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Prove that (f+g)(x) is an odd function, if f and g are odd functions (Stewart, Calculus)

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Suppose f(x) and g(x) are odd functions. Prove that (f+g)(x) is also an odd function.  Answer:  1. Strategy By definition, f is an odd function if and only if f(-x) = - f(x) To show (f+g) is an odd function, we need to show (f+g)(-x) = - (f+g)(x) 2. Explanation Since $f(x)$ and $g(x)$ are odd functions $\Rightarrow f(-x) =-f(x)$ and $g(-x) =-g(x)$ By definition of sum of functions. $(f+g)(-x) =f(-x)+g(-x)$ $=-f(x)-g(x)$ $=-(f(x)+g(x))$ $=-(f+g)(x)$ (by definition of sum of functions) $\Rightarrow(f+g)(-x) =-(f+g)(x)$ $\Rightarrow f+g$ is an odd function. Q.E.D. 
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