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

### 2 Answers by Expert Tutors

Jody B. | Undergraduate Math Tutor - 7+ years ExperienceUndergraduate Math Tutor - 7+ years Expe...
4.0 4.0 (1 lesson ratings) (1)
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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.
Parviz F. | Mathematics professor at Community CollegesMathematics professor at Community Colle...
4.8 4.8 (4 lesson ratings) (4)
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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.