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:

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

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

Re: Determine whether function is even or odd

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

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

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 $\displaystyle \displaystyle \begin{align*} f(-x) = (-x)^3\sin{(-x)} \end{align*}$.