How do you determine if #4x^5 / absx # is an even or odd function?

Question

Sophie Adams · Accepted Answer

The function is odd.

We have #f(x)=(4x^5)/|x|# We wish to find if #f# is even or odd. To make sure, let's remeber what those things mean: #{(f " is even" => f(-x)=f(x)),( f " is odd" => f(-x)=-f(x)) :}# Note that in both definitions we had #f(-x)# being compared to #f(x)#. Let's do the same. #f(x)=(4x^5)/|x|# #f(-x)=(4(-x)^5)/|-x|# For real numbers, #|a|=|-a|# by the definition of the absolute value. Now, because #x^5# has an odd exponent, #(-x)^5=-x^5#. For a general rule, if #k# is odd then #(-x)^k=-x^k# Hence, #f(-x)=-(4x^5)/|x|=-f(x)# We see now that #f# is an odd function.

Christian Antunez · Answer

undefined To determine if $ \frac{4x^5}{|x|} $ is an even or odd function, analyze its symmetry properties:

1. Even functions satisfy the condition $ f(-x) = f(x) $ for all $ x $ in their domain.
2. Odd functions satisfy the condition $ f(-x) = -f(x) $ for all $ x $ in their domain.

For $ \frac{4x^5}{|x|} $, notice that $ |x| $ is an even function because $ |x| = x $ when $ x \geq 0 $ and $ |x| = -x $ when $ x < 0 $. Therefore, $ \frac{4x^5}{|x|} $ is odd because when you substitute $ -x $ into the function, the absolute value will not change, but the sign of $ x $ will change, resulting in $ -\frac{4x^5}{|x|} $, which satisfies the condition for an odd function.