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How do you determine if # (x-1)^3(x-5)# is an even or odd function?

Answer 1

Benjamin Achtenberg

The given function is neither odd nor even.

If #f(-x)=f(x)# then the function is even,

but if #f(-x)=-f(x)# then the function is odd.

As #f(x)=(x-1)^3(x-5)#

#f(-x)=(-x-1)^3(-x-5)=(-(x+1))^3(-(x+5))# or

#f(-x)=(-(x+1)^3)(-(x+5))=(x+1)^3(x+5)#

Hence neither #f(-x)=f(x)# nor #f(-x)=-f(x)#

Hence the given function is neither odd nor even.

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Answer 2

Christian Antunez

To determine if a function is even, odd, or neither, you can use the properties of even and odd functions. An even function satisfies (f(x) = f(-x)) for all (x) in the domain, while an odd function satisfies (f(-x) = -f(x)).

Given the function (f(x) = (x-1)^3(x-5)), let's test if it's even or odd:

Substitute (-x) for (x):

[ f(-x) = ((-x)-1)^3((-x)-5) = (-x-1)^3(-x-5) ]

Simplify:

[ f(-x) = (-x-1)^3(-x-5) = (-(x+1))^3(-(x+5)) = -(x+1)^3(x+5) ]

This result, (f(-x) = -(x+1)^3(x+5)), does not equal (f(x) = (x-1)^3(x-5)) nor does it equal (-f(x)) due to the different terms inside the parentheses. Thus, (f(x)) 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.