anonymous
  • anonymous
Give an example of an even function and explain algebraically why it is even.
Mathematics
chestercat
  • chestercat
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ParthKohli
  • ParthKohli
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.
anonymous
  • anonymous
is that it?
ParthKohli
  • ParthKohli
Maybe.

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ParthKohli
  • ParthKohli
Do you know what \(|x|\) is?
anonymous
  • anonymous
x
ParthKohli
  • ParthKohli
And \(|-x|\).
anonymous
  • anonymous
x
ParthKohli
  • ParthKohli
Bang on!
ParthKohli
  • ParthKohli
And do you know what \((-x)^2\) is?
anonymous
  • anonymous
x?
ParthKohli
  • ParthKohli
Nope
anonymous
  • anonymous
o
anonymous
  • anonymous
x^2
ParthKohli
  • ParthKohli
Yes.
ParthKohli
  • ParthKohli
That's correct! =)
anonymous
  • anonymous
so how does that answer this question, bro? :)
ParthKohli
  • ParthKohli
You just answered the question, “explain algebraically why it is even”. Look back at the definition.
hba
  • hba
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
  • hba
@ParthKohli *
ParthKohli
  • ParthKohli
:)
ParthKohli
  • ParthKohli
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.

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