SOLUTION: Determine algebraically if the function f(x)= {{{ x-x^2sinx }}} is odd, even, or neither.

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Question 287718: Determine algebraically if the function f(x)= is odd, even, or neither.
Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
The function is even if:

f(x) = f(-x)

The function is odd if:

-f(x) = f(-x)

Here's a reference:

http://en.wikipedia.org/wiki/Even_and_odd_functions

Here's a graph of your function:

It's not even as far as I can tell.

It does look odd.

Only way to tell for sure is to give it the test.

First the even test.

Function is even if f(-x) = f(x)

f(x) = x - x^2 * sin(x)

f(-x) = (-x) - (-x)^2 * sin(-x) which becomes:

f(-x) = (-x) - (x^2 * (-sin(x))) which becomes:

f(-x) = -x + (x^2*sin(x))

Function is not even.

Next the odd test.

Function is odd if:

f(-x) = -f(x)

f(x) = x - x^2 * sin(x)

-f(x) = -(x - x^2 * sin(x)) which becomes:

-f(x) = -x + (x^2 * sin(x))

f(-x) from above became:

f(-x) = -x + (x^2 * sin(x)).

Looks like the function is odd.

We can test this out with numbers.

We know that the sine of 60 degrees = .866025404

60 degrees * pi/180 = 1.047197551 radians.

sine of 1.047197551 radians = .866025404

Since the sines are the same, we know we did the conversion from degrees to radians correctly.

When you graph the sine of a number, the x is assumed to be in radians.

That's why this conversion was necessary.

We will now take f(x) = x - x^2 * sin(x) and solve for f(1.047197551)

f(1.047197551) = 1.047197551 - (1.047197551)^2 * sin(1.047197551).

That becomes f(1.047197551) = .097494425

Now we want to find f(-x) when x = 1.047197551.

f(-1.047197551) = (-1.047197551) - (-1.047197551)^2 * sin(-1.047197551).

That becomes f(-1.047197551) = -.097494425

f(x) = .097494425
-f(x) = -.097494425
f(-x) = -.097494425

-f(x) = f(-x) confirming that the function is odd.

When you look at the graph, you will see that it looks symmetric except that the signs are reversed.