Question

How do you determine if `f(x)= (-x³)/(3x²-4)` is an even or odd function?

Answer 1

See below

Explanation:

_ GENERALLY SPEAKING _

There are tests:

EVEN function: f(-x) = f(x)

so, for example, the ubiquitous cosine function is even as `cos (-x) = cos x`. Generally speaking, even functions reflect across the y-axis.

ODD function: f(-x) = - f(x)

the equally ubiquitous sine function is odd as `sin(- x) = - sin x`. Odd functions show symmetry in terms of a 180 deg rotation about the origin. But they don't have to pass through the origin, eg `f(x) = 1/x, f(-x) = - 1/x = - f(x)` is odd.

NB a function can be neither odd or even. The `ln x` and `e^x` functions, for example.

_ THE ACTUAL QUESTION _

so here we test `f(x)= (-x³)/(3x²-4)` by looking at `f(-x)`

which is

`f(-x) = (-(-x)³)/(3(-x)²-4)`

`= (x³)/(3x²-4) = - f(x)`

this is odd

_ A GRAPH _

graph{ (-x^3)/(3x^2-4) [-18.02, 18.02, -9.01, 9.01]}