How do you determine if #f(x) = 6x^5 -5x# is an even or odd function?

Question

Elijah Allen · Accepted Answer

#f(x)# is odd.

an odd function is defined as: #f(-x)=-f(x)# an even function is defined as: #f(-x)=f(x)# #f(x)=6x^5-5x# #f(-x)=6(-1)^5-5(-x)# #f(-x)=-6x+5(x)# #f(-x)=-(6x^5-5x)=-f(x)# #:.#odd

Madelyn Anderson · Answer

undefined To determine if a function is even or odd, we need to check its symmetry.

1. **Even function:** A function $ f(x) $ is even if $ f(-x) = f(x) $ for all $ x $ in the domain. In other words, if the function is symmetric about the y-axis.

2. **Odd function:** A function $ f(x) $ is odd if $ f(-x) = -f(x) $ for all $ x $ in the domain. In other words, if the function is symmetric about the origin.

For the function $ f(x) = 6x^5 - 5x $:

1. **Even or odd?** Let's first check if it is even or odd. We evaluate $ f(-x) $:

$ f(-x) = 6(-x)^5 - 5(-x) $
   
   $ f(-x) = -6x^5 + 5x $

2. **Check for even:** If $ f(x) = f(-x) $, the function is even. However, $ f(x) \neq f(-x) $, so the function is not even.

3. **Check for odd:** If $ f(-x) = -f(x) $, the function is odd. From our earlier evaluation, $ f(-x) = -6x^5 + 5x $, and $ -f(x) = -6x^5 + 5x $. Since $ f(-x) = -f(x) $, the function is odd.

Therefore, $ f(x) = 6x^5 - 5x $ is an odd function.