# Determine whether function is even or odd

• May 9th 2013, 04:26 PM
Unreal
Determine whether function is even or odd
Hi!

Determine whether the function is even, odd or neither.

1. $\displaystyle f(x) = x^3 \sin x$

The function is even since:

$\displaystyle f(-x) = -x^3\sin(-x) = x^3sinx$

The result of cubing a negative is negative and taking the sine of a negative gives a negative, and multiplying two negatives gives a positive.
Correct any flaws in this thinking process please!

2. $\displaystyle f(x) = \cos(\sinx)$

This function is even since:

$\displaystyle f(-x) = \cos(\sin(-x) = \cos(-\sin(x)) = -\cos(\sin(x) = \cos(\sinx)$
• May 9th 2013, 05:27 PM
topsquark
Re: Determine whether function is even or odd
Quote:

Originally Posted by Unreal
Hi!

Determine whether the function is even, odd or neither.

1. $\displaystyle f(x) = x^3 \sin x$

The function is even since:

$\displaystyle f(-x) = -x^3\sin(-x) = x^3sinx$

The result of cubing a negative is negative and taking the sine of a negative gives a negative, and multiplying two negatives gives a positive.
Correct any flaws in this thinking process please!

2. $\displaystyle f(x) = \cos(\sinx)$

This function is even since:

$\displaystyle f(-x) = \cos(\sin(-x) = \cos(-\sin(x)) = -\cos(\sin(x) = \cos(\sinx)$

The first is correct. For the second, recall that cos(-x) = cos(x). Cosine is an even function, not odd.

-Dan
• May 9th 2013, 05:39 PM
Unreal
Re: Determine whether function is even or odd
$\displaystyle f(-x) = \cos(\sin(-x) = \cos(-\sin(x)) = -\cos(\sin(x) = \cos(sinx)$
• May 9th 2013, 05:43 PM
topsquark
Re: Determine whether function is even or odd
Quote:

Originally Posted by Unreal
$\displaystyle f(-x) = \cos(\sin(-x) = \cos(-\sin(x)) = -\cos(\sin(x) = \cos(sinx)$

$\displaystyle f(-x) = cos( sin( -x )) = cos( -sin(x) ) = cos( sin(x) ) = f(x)$

Thus f(x) is even.

-Dan
• May 9th 2013, 09:27 PM
Prove It
Re: Determine whether function is even or odd
Quote:

Originally Posted by topsquark
The first is correct. For the second, recall that cos(-x) = cos(x). Cosine is an even function, not odd.

-Dan

The first is NOT correct (or at least is incomplete), as the OP should have written \displaystyle \displaystyle \begin{align*} f(-x) = (-x)^3\sin{(-x)} \end{align*}.