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

Answer 1

An even function is a function where #f(-x)=f(x)# and an odd function is a function where #f(-x)=-f(x)#

Take a look at this #cos# graph: graph{cos(x) [-10, 10, -5, 5]} As you can see, the #y#-value for #pi/2# is the same as the #y#-value for #-pi/2#, so it's an even function.

Now look at this #sin# graph: graph{sin(x) [-10, 10, -5, 5]}

The #y#-value for #pi/2# is #1# whilst the #y#-value for #-pi/2# is #-1#, so it's an odd function.

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Answer 2

Cora Alexander

To determine if ( f(x) = x - |x| ) is an even or odd function:

Even function: ( f(x) = f(-x) ) for all ( x ) in the domain.
Odd function: ( f(x) = -f(-x) ) for all ( x ) in the domain.

For the given function:

( f(x) = x - |x| )

Replace ( x ) with ( -x ):

( f(-x) = -x - |-x| )

Simplify:

( f(-x) = -x - |x| )

Compare ( f(-x) ) with ( f(x) ):

( f(-x) = -f(x) )

Since ( f(-x) = -f(x) ), the function is 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.