How do you know if #f(x) = 4x^2 + x^4 -120# is an even or odd function?

Question

Anna Allen · Accepted Answer

This function is even, because all the exponents of #x# are even

This is not a general rule, but it is enough for polynomials. If all exponents of #x# are even, then the whole function is even, on the other hand if all exponents are odd, then the whole function is also odd. For other function the rule is not so simple. We have to calculate #f(-x)#. If we get #f(-x)=f(x)#, then the function is even, else if we get: #f(-x)=-f(x)# then the function is odd, else the function is neither even or odd. For the given function we get: #f(-x)=4*(-x)^2+(-x)^4-120# #f(-x)=4x^2+x^4-120# #f(-x)=f(x)# The function is even.

Elijah Allen · Answer

undefined To determine if a function is even or odd, we evaluate whether it satisfies the properties of even or odd functions.

1. Even Function:
   f(x) is even if f(-x) = f(x) for all x in the domain.

2. Odd Function:
   f(x) is odd if f(-x) = -f(x) for all x in the domain.

For the function f(x) = 4x^2 + x^4 - 120:

1. Even Function Test:
   f(-x) = 4(-x)^2 + (-x)^4 - 120
         = 4x^2 + x^4 - 120 (same as f(x))

Since f(-x) = f(x), the function satisfies the condition for even functions.

2. Odd Function Test:
   f(-x) = 4(-x)^2 + (-x)^4 - 120
         = 4x^2 + x^4 - 120

However, for the function to be odd, we need f(-x) = -f(x).

-f(x) = -[4x^2 + x^4 - 120]
         = -4x^2 - x^4 + 120

Since f(-x) is not equal to -f(x), the function does not satisfy the condition for odd functions.

Therefore, the function f(x) = 4x^2 + x^4 - 120 is an even function.