How do you determine if #f(x) = - 4x^5 - 6x^3# is an even or odd function?

Question

Caleb Alexander · Accepted Answer

Odd, see below.

It's an odd function right out of the box, because the exponents are odd. But we can apply the tests: odd : #f(-x) = - f(x)# even : #f(-x) = f(x)# Here: #f(-x) = - 4(-x)^5 - 6(-x)^3 # # = 4x^5 + 6x^3# #= - f(x)#

Sadie Ackerman · Answer

undefined To determine if $ f(x) = - 4x^5 - 6x^3 $ is an even or odd function, we analyze its symmetry:

1. Even functions satisfy $ f(x) = f(-x) $. To test this, substitute $-x$ for $x$ in the function and simplify.
2. Odd functions satisfy $ f(x) = -f(-x) $. To test this, negate the function and substitute $-x$ for $x$ in the negated function. If the result is equal to the original function, it's odd.

Applying these tests to $ f(x) = - 4x^5 - 6x^3 $:

Even test: $ f(-x) = - 4(-x)^5 - 6(-x)^3 = - 4(-x^5) - 6(-x^3) = - 4x^5 + 6x^3 $ which is not equal to $ f(x) $. Therefore, $ f(x) $ is not even.

Odd test: Negate the function $ -(- 4x^5 - 6x^3) = 4x^5 + 6x^3 $ and substitute $-x$ for $x$: $ 4(-x)^5 + 6(-x)^3 = 4(-x^5) + 6(-x^3) = - 4x^5 - 6x^3 $. Since this is equal to $ f(x) $, $ f(x) $ is an odd function.

Sadie Ackerman · Answer

undefined To determine if a function is even, odd, or neither, you evaluate whether it satisfies the following conditions:

1. Even Function: f(x) = f(-x) for all x in the function's domain.
2. Odd Function: f(x) = -f(-x) for all x in the function's domain.

For the function f(x) = -4x^5 - 6x^3:

1. To check for evenness, substitute -x for x and simplify:
   f(-x) = -4(-x)^5 - 6(-x)^3
         = -4(-x^5) - 6(-x^3)
         = -4x^5 - 6x^3
   Since f(-x) = f(x), the function is even.

2. To check for oddness, substitute -x for x and simplify:
   -f(-x) = -(-4(-x)^5 - 6(-x)^3)
          = -(-4(-x^5) - 6(-x^3))
          = -(-4x^5 + 6x^3)
          = 4x^5 - 6x^3
   Since -f(-x) is not equal to f(x), the function is not odd.

Therefore, the function f(x) = -4x^5 - 6x^3 is an even function.