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Determine whether the function f(x) = -3x^3+4x is even, odd, or neither.

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2 Answers

The definition of a odd function is f(-x)=-f(x) & the definition of an even function is f(-x)=f(x). All other cases are considered neither. 

So lets test your function now. F(-x)= -3 (-x)3+4(-x)= 3x3-4x= -F(x). Therefore it is an odd function. 

What does it mean??

This means if you check the input value for a odd function, lets say F(5), and has some value A. If you put in F(-5) you will get -A. So there is a symmetry about odd function. This is similar for even functions except you will find it would A again, so it has a different kind of symmetry.    

Replace the x with -x.  If you get the same function back with no sign changes, the function is even.  If you get the function with every sign reversed, it's odd.  In all other cases, it's neither.  For example, suppose you had f(x) = x3.  Replace the x with -x and you get f(x) = (-x)3 = -x * -x * -x = -x3 so the function would be odd.  Now try it on your problem.