How do you determine if # abs (x)/ x# is an even or odd function?

Question

Sadie Ackerman · Accepted Answer

x = 0 is a point of graph-breaking-discontinuity. For x > o, y = #|x|/x = x/x = 1# and for x < 0, y = #-x/x=-1#. Sans x= 0, y is an odd function.

Answer is brief but self-explanatory.

Chloe Alexander · Answer

we know 
#|x|=-x# when #x<0 #
and#|x|=x# when #x>0 #
so for  #f(x)=|x|/x#
putting #x=-x# #f(-x)=|-x|/-x=-|x|/x=-f(x)# so it is an odd function

Greyson Brown · Answer

undefined To determine if $ \frac{\left| x \right|}{x} $ is an even or odd function, we examine its symmetry properties with respect to the origin.

1. Even Function:
A function $ f(x) $ is even if $ f(-x) = f(x) $ for all $ x $ in its domain. In other words, if replacing $ x $ with $ -x $ doesn't change the function's value.
$$ \frac{\left| -x \right|}{-x} = \frac{\left| x \right|}{x} $$

2. Odd Function:
A function $ f(x) $ is odd if $ f(-x) = -f(x) $ for all $ x $ in its domain. In other words, if replacing $ x $ with $ -x $ changes the sign of the function's value.
$$ \frac{\left| -x \right|}{-x} = -\frac{\left| x \right|}{x} $$

Now, let's evaluate $ \frac{\left| x \right|}{x} $ for both cases:

When $ x $ is positive, $ \frac{\left| x \right|}{x} = \frac{x}{x} = 1 $.
When $ x $ is negative, $ \frac{\left| x \right|}{x} = \frac{-x}{x} = -1 $.

Since $ \frac{\left| x \right|}{x} $ changes sign when $ x $ changes sign, it is an odd function.