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Even and odd functions: Find the mistake

See another student's work when trying to determine whether a function is even, odd, or either, and decide whether they made a mistake, and if so, where.

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• if f(x) = x - ∛𝑥
and f(-x) = -x + ∛𝑥
shouldn't they be equivalent because adding a negative and subtracting a positive give the same result?
• Well, if you want to test something like this, you can always plug in arbitrary numbers and see if it works. Let's use 8:
8 - ∛(8) = 8 - 2 = 6
-8 + ∛(8) = -8 + 2 = -6
6 ≠ -6
The cube root of x is a completely different value than just x, so you can't really just switch the negative sign as if you were doing x - x and -x + x.
• if f(-x) = -f(-x) determines an odd function, then why does f(-x)= -f(x) makes sense?
• f(-x)=-f(-x) determines the zero function, not just any odd function. Odd function are defined as satisfying f(-x)=-f(x).
• Can someone explain what Sal did from to ?
• ah I dislike these kind of questions, it’s like find the mistake yourself don’t make me find it lol (no offense meant, Sal is great)
• Sal made a mistake on step two as well. He dropped the negative sign on the sign on the x before he distributed the negative sign at , and he wound up with the wrong expression for -f(-x). -f(-x) = f(x), but Sal ha the x as -x because he dropped a sign before distributing the negative.
(1 vote)
• What Sal wants to do is to check if 𝑓(−𝑥) = −𝑓(𝑥)

He already figured out that 𝑓(−𝑥) = −𝑥 + ∛𝑥

He also knows that 𝑓(𝑥) = 𝑥 − ∛𝑥
Thereby, −𝑓(𝑥) = −(𝑥 − ∛𝑥) = −𝑥 + ∛𝑥 = 𝑓(−𝑥)

There's no mistake.
• Shouldn't Jayden have found -f(-x) instead of -f(x), and that was his mistake? So we shouldn't bother about his calculation for -f(x) at all, because he should have tried to find -f(-x) in the first place.