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

The given function is neither odd nor even.

Hence the given function is neither odd nor even.

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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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