How do you know if #f(x)=2x^4+3x^2# is an even or odd function?

Question

Lincoln Anderson · Accepted Answer

This function is even because #f(-x) = f(x)# #f(x)=2x^4+3x^2# #f(-x)=2(-x)^4+3(-x)^2=2x^4+3x^2 = f(x)# [ NOTE : If the function is odd , then #f(-x)=-f(x)# ] hope that helped

Greyson Brown · Answer

undefined To determine whether a function $ f(x) = 2x^4 + 3x^2 $ is even, odd, or neither:

1. **Even Function:**
   A function is even if $ f(-x) = f(x) $ for all $ x $ in its domain. To test if $ f(x) $ is even, substitute $ -x $ for $ x $ in the function and simplify. If the resulting expression is identical to the original function, then the function is even.

2. **Odd Function:**
   A function is odd if $ f(-x) = -f(x) $ for all $ x $ in its domain. To test if $ f(x) $ is odd, substitute $ -x $ for $ x $ in the function and simplify. If the resulting expression is the negative of the original function, then the function is odd.

For the function $ f(x) = 2x^4 + 3x^2 $:

1. Test for evenness:
   $ f(-x) = 2(-x)^4 + 3(-x)^2 = 2x^4 + 3x^2 $

Since $ f(-x) = f(x) $, the function is even.

2. Test for oddness:
   $ f(-x) = 2(-x)^4 + 3(-x)^2 = 2x^4 + 3x^2 $

Since $ f(-x) $ is not equal to $ -f(x) $, the function is not odd.

Therefore, the function $ f(x) = 2x^4 + 3x^2 $ is even.