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

-
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. 
Post a Comment

Proof: f + g is an even function, if f and g are even functions. (Stewart, Calculus)

-

Question: Suppose f(x) and g(x) are even functions. Prove that (f+g)(x) is also an even function. 


Answer:

1. Strategy

Definition: a function h(x) is an even function if h(x) = h(-x)

To prove (f+g)(x) is an even function, we need to show (f+g)(x) = (f+g)(-x) 


2. Explanation 

Recall $(f+g)(x)=f(x)+g(x)$ (by definition of sum of functions)

Since $f(x)$ and $g(x)$ are even functions

$\Rightarrow f(x)=f(-x)$ and $g(x) = g(-x)$

$\Rightarrow(f+g)(x)=f(x)+g(x)=f(-x)+g(-x)=(f+g)(-x)$

$\Rightarrow(f+g)(x)=(f+g)(-x)$

$\Rightarrow(f+g)$ is an even function. Q.E.D. 


Comments

Post a Comment

Popular posts from this blog

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

-
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. 
Read more

[Math] A proof of Triangle Inequality (1)

-
Prove: $|x+y| \leq|x|+|y|$ (triangle inequality)  $x y \leq|x y| $  $\Leftrightarrow  x y \leq|x||y| $  $\Leftrightarrow  2 x y \leq 2|x||y| $ $\Leftrightarrow  x^2+y^2+2 x y \leq x^2+y^2+2|x||y| $  $\Leftrightarrow x^2+y^2+2 x y \leq\left.| x\right|^2+|y|^2+2|x||y| $  $\Leftrightarrow  (x+y)^2 \leq(|x|+|y|)^2$ [equation 1] case 1) if $x+y>0$ then $ |x+y| = x+y$  [equation1] $ \Rightarrow (|x+y|)^2 \leq(|x|+|y|)^2 $ [equation 2] $ \Leftrightarrow |x+y| \leq|x|+|y| $  case 2) if $x+y<0$ then $ |x+y| = -(x+y)$  note that $(-(x+y))^2 = (x+y)^2 = |x+y|^2$ $\Rightarrow$ [equation 2] $\Rightarrow$ $|x+y| \leq|x|+|y|$ $\blacksquare$ Published on: January 10, 2023  By: ComputeFinance
Read more

[Python] Sublime Text: set a build on Python 2.7 creating a sublime-build file (solution to WinError 2 problem)

-
Image
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
Read more