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### Course: Integrated math 3 > Unit 6

Lesson 3: Symmetry of functions- Function symmetry introduction
- Function symmetry introduction
- Even and odd functions: Graphs
- Even and odd functions: Tables
- Even and odd functions: Graphs and tables
- Even and odd functions: Equations
- Even and odd functions: Find the mistake
- Even & odd functions: Equations
- Symmetry of polynomials

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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.

## Want to join the conversation?

- 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?(6 votes)- 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.(6 votes)

- if f(-x) = -f(-x) determines an odd function, then why does f(-x)= -f(x) makes sense?(5 votes)
- f(-x)=-f(-x) determines the zero function, not just any odd function. Odd function are defined as satisfying f(-x)=-f(x).(5 votes)

- Can someone explain what Sal did from1:20to1:34?

Thanks in advance!(6 votes) - 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)(4 votes)
- 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 at2:36, 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.(6 votes)

- 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.(2 votes)
- i got

-x/1+x To be even, it would have to go back to

x/1+x To be odd, it would have to be

-(x/1+x)(2 votes) - Why does step 2 say to check whether f(-x) is equal to f(-x). Shouldn't it be -f(x)?(1 vote)
- I can't speak to the original writer's motivations, but f times -x would be the same as -f times x, since one negative times one positive always makes a negative. It's probably just a formatting typo, but it doesn't seem to have any bearing on the equation.(1 vote)

## Video transcript

- [Instructor] We are told
Jayden was asked to determine whether f of x is equal to
x minus the cube root of x is even, odd, or neither. Here is his work. Is Jayden's work correct? If not, what is the first step
where Jayden made a mistake? So pause this video and
review Jayden's work, and see if it's correct, or if it's not correct tell
me where it's not correct. All right, now let's work this together. So, let's see, just to remind ourselves what Jayden's trying to
do, he's trying to decide, whether f of x is even, odd, or neither. And f of x is expressed, or is defined, as x minus the cube root of x. So let's see, the first
thing that Jayden did is he's trying to figure out
what is f of negative x? Because remember, if f of negative x is equal to f of x, we are even, and if f of negative x is equal to negative f
of x, then we are odd. So it makes sense for him to find the expression
for f of negative x. So he tries to evaluate f of negative x, and when he does that, everywhere where he sees an x in f of x, he replaces it with a negative x. So that seems good. And then, let's see, this
becomes a negative x, that makes sense, minus, and then, a negative x under the radical, and this is a cube root right over here, that's the same thing
as negative one times x. The cube root of negative
one is negative one. So he takes that negative
out of the radical, out of the cube root. So this makes sense, and
so then he has a negative x and you subtract a negative,
you get a positive. So then that makes sense. And then, the next thing he
says is, or he's trying to do, is check if f of negative x is equal to f of x or f of negative x. So he's gonna check whether
this is equal to one of them. And so here Jayden says, negative
x plus the cube root of x, so that's what f of negative
x, what he evaluated it to be, isn't the same as f of x, now
let's see is that the case? Is it not the same as f of x? Yup it's definitely, it's
not the same as f of x, or negative f of x which is equal to negative x minus the cube root of x. Now that seems a little bit fishy. Did he do the right
thing, right over here? Is negative f of x equal to negative x minus the cube root of x? Let's see, negative of f of x is going to be a negative
times this entire expression, it's going to be a negative up front, times x minus the cube root of x, and so this is going to be equal to, you distribute the negative sign, you get negative x plus
the cube root of x. So Jayden calculated the wrong negative f of x right over here. So, he isn't right that negative
x plus the cube root of x, it is actually the same
as negative f of x. So he's wrong right over here. So Jayden's mistake is right
over here, really it looks like he didn't evaluate
negative f of x correctly. So Jayden's work, is
Jayden's work correct? No. If not, what is the first step
where Jayden made a mistake? Well it would be step two. What he should have said is, it actually is the same
as negative f of x, and so therefore his conclusion should be that f of x is odd.