mitchelsewbaran

Give an example of an even function and explain algebraically why it is even.

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ParthKohli

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So, an even function \(f\) is a function where for all \(x\), \(f(x) = f(-x)\). The square function and the absolute-value functions are pretty cool examples.

mitchelsewbaran

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is that it?

ParthKohli

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Maybe.

ParthKohli

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Do you know what \(|x|\) is?

ParthKohli

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And \(|-x|\).

ParthKohli

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Bang on!

ParthKohli

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And do you know what \((-x)^2\) is?

mitchelsewbaran

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x?

ParthKohli

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Nope

mitchelsewbaran

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x^2

ParthKohli

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Yes.

ParthKohli

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That's correct! =)

mitchelsewbaran

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so how does that answer this question, bro? :)

ParthKohli

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You just answered the question, “explain algebraically why it is even”.
Look back at the definition.

hba

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If
f(-a)=f(a) then it is even
example
f(x)=x^2
f(-x)=(-x)^2
f(-x)=x^2
therfore,
f(x)=f(-x)
@parth explained it well :)

hba

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@ParthKohli *

ParthKohli

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:)

ParthKohli

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Let me do that for an absolute-value function too! So we define the absolute value function as follows\[f(x) = \cases{x \ \ \text{iff} \ \ x >0 \\ -x \ \ \text{iff} \ \ x < 0 }\]It is clear that \(f(-x)=f(x)=x\), hence an even function.