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

1 Answer
Eddie
Aug 8, 2016

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]}

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