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How do you determine if #f(x)=6x-root3x# is an even or odd function?

Answer 1

Christian Antunez

This #f(x)# is an odd function since #f(-x) = -f(x)# for any #x#.

#f(x)# is an even function if #f(-x) = f(x)# for any #x#

#f(x)# is an odd function if #f(-x) = -f(x)# for any #x#

For any value of #x# we have:

#f(-x) = 6(-x)-root(3)(-x) = -6x+root(3)(x) = -(6x-root(3)(x)) = -f(x)#

So #f(x)# is an odd function.

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Answer 2

Xavier Alonso

To determine if the function ( f(x) = 6x - \sqrt{3x} ) is even or odd, evaluate ( f(-x) ) and compare it with ( f(x) ):

Even function: If ( f(-x) = f(x) ) for all ( x ) in the domain of the function, then the function is even.
Odd function: If ( f(-x) = -f(x) ) for all ( x ) in the domain of the function, then the function is odd.

Let's evaluate ( f(-x) ) and compare:

[ f(-x) = 6(-x) - \sqrt{3(-x)} = -6x - \sqrt{-3x} ]

Now, compare ( f(-x) ) with ( f(x) ):

[ f(-x) \neq f(x) ]

Also,

[ f(-x) \neq -f(x) ]

Since neither condition holds, the function ( f(x) = 6x - \sqrt{3x} ) is neither even nor odd.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.