## mitchelsewbaran 2 years ago Give an example of an even function and explain algebraically why it is even.

1. 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.

2. mitchelsewbaran

is that it?

3. ParthKohli

Maybe.

4. ParthKohli

Do you know what $$|x|$$ is?

5. mitchelsewbaran

x

6. ParthKohli

And $$|-x|$$.

7. mitchelsewbaran

x

8. ParthKohli

Bang on!

9. ParthKohli

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

10. mitchelsewbaran

x?

11. ParthKohli

Nope

12. mitchelsewbaran

o

13. mitchelsewbaran

x^2

14. ParthKohli

Yes.

15. ParthKohli

That's correct! =)

16. mitchelsewbaran

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

17. ParthKohli

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

18. 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 :)

19. hba

@ParthKohli *

20. ParthKohli

:)

21. 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.