functions help please

Question

Let f(x)=x^(3 )-7x. Determine if this function is even, odd or neither. Explain/Show work.

 

Answer 1

A function is even if f(-x) = f(x), while it is odd if f(-x) = -1*f(x)... These equations have to be true for all x.

If f(-x) does not equal either f(x) or -1*f(x), then it can't be called either even or odd.

 

So, let's evaluate f(-x):

 

(-x)^3 - 7*(-x) = (-1)^3*(x)^3 - (-1)*7*x

                       = (-1)* (x^3 - 7x)  

...because (-1)*3 = -1, and that would leave a -1 that could be factored out of the subtraction...

 

Since f(x) = (x^3-7x), we have just shown that f(-x) = -1*f(x), and we can conclude that f(x) is odd.

Answer 2

If f( x ) = f ( -x)  , then the function is even function.

 

If f( x) = - f ( -x) , then the function is odd.

 

  f( x ) = X^3 - 7 X

 

 f ( -x ) =( -x ) ^3 - 7 ( -x )= - X^3 + 7x  = - ( X^3 - 7X)  , therefore it is an odd function.

 

  f( x ) ≠f(- x)   , therefore f( x) is not an even function.