Function parity problem

[Ognjen]

Problem formulation:

If function f is odd with period 8 and if f(x) = x^4 - 16x^2, for x belonging to the interval [0,4], then what does f(198) equal to ?

The text clearly elicits assignment of the following equation as the starting step: f(-x) = f(x) ( since the function is said to be odd ).

However, the official solution now says that from the foregoing equation, it can be deduced that f(x) = -f(x). How they mathematically come from f(-x) = f(x) to this, is beyond me.

It now says the following: If x is an element of the interval [-4,0], it follows that -x is an element of the interval [0,4].

I don't understand how they come to x being an element of [-4,0] at the first place ! I would conclude that -x is an element of [-4,0] ( opposite sign of the domain that stands for x, obviously ).

Now, based on the equation of parity ( the basic one, f(-x) = f(x) ) and the f(x) definition given in the text of the problem, we have:

[math]f(x) = -((x)^4 - 16(-x)^2) = -x^4 + 16x^2[/math]
Now comes the most confusing part. I will cite the official solution.

On the interval [-4,4], function f is thus defined with:

f(x) = { x^4 + 16x^2, x element of [0,4]
{ -x^4 + 16x^2, x element of [-4,0]

( the beginning braces should be fused into one, designating ''branching'' of the same function )

Regardless of whether -x is of the interval I expressed confusion about or not, the first equation just cannot make sense to me.

Whether the input x is positive or negative, since the 2 variables in the function definition are put to power of an even number, the equation CANNOT change in sign, and thus has to be even. This doesn't make sense even considering the beginning premises of the problem !

Either way, if we ignore that for some reason, we can only have

[math]f(x) = x^4 - 16x^2[/math]
or

[math]f(x) = -x^4 + 16x^2[/math]
( depending on whether we just repeat the definition of f(x) or use the equation f(x) = -f(x) )



I do notice my reason has to be wrong as it leads to refutation of the initial premises ( and ultimately ends nowhere ), but I don't exactly understand where and how !



I would be immensely thankful if anybody could help me with clarification of this issue. Thank you in advance.

 

[Dr.Peterson]

The text clearly elicits assignment of the following equation as the starting step: f(-x) = f(x) ( since the function is said to be odd ).

However, the official solution now says that from the foregoing equation, it can be deduced that f(x) = -f(x). How they mathematically come from f(-x) = f(x) to this, is beyond me.

It now says the following: If x is an element of the interval [-4,0], it follows that -x is an element of the interval [0,4].

I don't understand how they come to x being an element of [-4,0] at the first place ! I would conclude that -x is an element of [-4,0] ( opposite sign of the domain that stands for x, obviously ).

You seem to misunderstand how negatives work.

If x is in [0,4], then -x is a negative number, in [-4,0]. But if x is in [-4,0], it is a negative number, and -x is a positive number, in [0,4]. Do you understand that?

I hope you mistyped when you said f(-x) = f(x); for an odd function f(-x) = -f(x), not f(x)! If they said what you show, then not only are they wrong, but you are wrong in agreeing. So I'll assume it said the right thing.

But in that case, you should have no objection to f(x) = -f(x), which is clearly equivalent.

I think you have other issues, but I can't be sure until we clear this up. Please correct what you wrote, or else move on with the statement corrected.

And, while you're at it, it would be good to learn from past discussions, and show us the entire problem and solution you are asking about, so we can be sure exactly what you are seeing and not get (too) confused. If you can provide these in a form we can pass through Google Translate, or do it for us yourself, it will be even better (though I'd still want to see the original so we can be sure you copied what it says).

 

[JeffM]

[math]f(x) \text { is even if } f(x) = f(-x), \text { and}\\ f(x) \text { is odd if } f(x) = - f(-x).[/math]
So we start with the fact that you have confused the definitions of odd and even functions.

And I think they are deliberately being a bit tricky in the way the formula for fx) seems to have been stated. If, for all x, [imath]f(x) = x^4 - 16x^2[/imath], then f(x) is an even function rather than an odd function. So the general formula for f(x) can be simplified to [imath]x^4 - 16x^2[/imath] in the interval [0, 4], but that cannot be the general formula for f(x) for all values of x, a result that you can reach independently because f(x) is said to be periodic.

 

[Steven G]

If function f is odd with period 8 and if f(x) = x^4 - 16x^2, for x belonging to the interval [0,4], then what does f(198) equal to ?
As JeffM pointed out, f is NOT an odd function. Personally I would stop there.

 

[Dr.Peterson]

If function f is odd with period 8 and if f(x) = x^4 - 16x^2, for x belonging to the interval [0,4], then what does f(198) equal to ?
As JeffM pointed out, f is NOT an odd function. Personally I would stop there.

Not at all! We are told only that f(x) is given by that expression over that interval, and it can be extended to the rest of the number line by using its parity and its period. We are not told that f is given by that expression everywhere!

Oddness lets you extend from [0,4] to [-4,4], and then the period lets you extend to other intervals.

 

[Ognjen]

You seem to misunderstand how negatives work.

If x is in [0,4], then -x is a negative number, in [-4,0]. But if x is in [-4,0], it is a negative number, and -x is a positive number, in [0,4]. Do you understand that?

I hope you mistyped when you said f(-x) = f(x); for an odd function f(-x) = -f(x), not f(x)! If they said what you show, then not only are they wrong, but you are wrong in agreeing. So I'll assume it said the right thing.

But in that case, you should have no objection to f(x) = -f(x), which is clearly equivalent.

I think you have other issues, but I can't be sure until we clear this up. Please correct what you wrote, or else move on with the statement corrected.

And, while you're at it, it would be good to learn from past discussions, and show us the entire problem and solution you are asking about, so we can be sure exactly what you are seeing and not get (too) confused. If you can provide these in a form we can pass through Google Translate, or do it for us yourself, it will be even better (though I'd still want to see the original so we can be sure you copied what it says).

I'm sorry for angering you with my lack of knowledge. I provided the images of the formulation of the problem ( the number of the problem is 22. ) and the official solution of the given problem ( in original ).

Yes, I mistyped the parity equation, sorry for that.

I don't understand the value interval. If x is in [0,4], then it simply cannot be in [-4,0] no matter what, right ? So I don't understand the meaning of your ''but if x is in [-4,0]'' when it cannot be there according to the formulation of the problem. I understand that for some reason I must be wrong about this, but I don't understand why or how...

 

[Dr.Peterson]

I'm sorry for angering you with my lack of knowledge.

Angry? No, but maybe irritated that by not showing the original, and mistyping it, you left me quite confused at first, and unsure where your real difficulties lie.

I don't understand the value interval. If x is in [0,4], then it simply cannot be in [-4,0] no matter what, right ? So I don't understand the meaning of your ''but if x is in [-4,0]'' when it cannot be there according to the formulation of the problem. I understand that for some reason I must be wrong about this, but I don't understand why or how...

No one is saying that if x is in [0,4], then the same x is in [-4,0]; or that x must always be in [0,4]. I said this:

If x is in [0,4], then -x is a negative number, in [-4,0]. But if x is in [-4,0], it is a negative number, and -x is a positive number, in [0,4]. Do you understand that?

Take an example. Suppose x is 3. Then -x is -3. And -3 is in [-4,0].

If, instead, we choose for x to be -3, which is in [-4,0], then -x is +3, which is in [0,4]. And since that is true, we can use the formula to find f(-x), and therefore f(x).

It now says the following: If x is an element of the interval [-4,0], it follows that -x is an element of the interval [0,4].

I don't understand how they come to x being an element of [-4,0] at the first place ! I would conclude that -x is an element of [-4,0] ( opposite sign of the domain that stands for x, obviously ).

So if we choose a value of x in [-4,0], such as my -3, then -x is in [0,4].

The problem tells you how to calculate f(x) when x is in [0,4]. The goal now is to figure out how to find f(x) when x is not in that interval, extending the function by using its oddness, and then its parity. So at this point, they are saying, let's pick a value of x that is in [-4,0], and see if we can find f(x).

I think you may be making a mistake similar to the one Stephen appears to have made, misreading the problem. It doesn't say that x can only be in [0,4]; then they couldn't ask for f(198) at all, could they?

It tells you how f is defined for some values of x, and gives you additional information about how the function behaves for other values (parity and period), which you can use to extend the function to other values.

It will end up looking something like this:

​

The red part is what they gave you, from 0 to 4. The green part is what you get when you apply oddness to that piece. The purple part is what you get when you use the period to extend to all real numbers. In the end, x can be any real number.

 

[JeffM]

I was responding when Dr. Peterson did respond. Having read his response, I have nothing left to add.

 

[Steven G]

Not at all! We are told only that f(x) is given by that expression over that interval, and it can be extended to the rest of the number line by using its parity and its period. We are not told that f is given by that expression everywhere!

Oddness lets you extend from [0,4] to [-4,4], and then the period lets you extend to other intervals.

Dr P,
I am a bit confused so am asking that you explain things to me.
It seems that you are saying that one period of f is from [0,4]. If you are saying that, I don't see that.
On the interval [0,4] f, as far as I can see, is an even function. I think that you are saying that it is odd(??) How can that be??

 

[Ognjen]

Angry? No, but maybe irritated that by not showing the original, and mistyping it, you left me quite confused at first, and unsure where your real difficulties lie.


No one is saying that if x is in [0,4], then the same x is in [-4,0]; or that x must always be in [0,4]. I said this:

Take an example. Suppose x is 3. Then -x is -3. And -3 is in [-4,0].

If, instead, we choose for x to be -3, which is in [-4,0], then -x is +3, which is in [0,4]. And since that is true, we can use the formula to find f(-x), and therefore f(x).

So if we choose a value of x in [-4,0], such as my -3, then -x is in [0,4].

The problem tells you how to calculate f(x) when x is in [0,4]. The goal now is to figure out how to find f(x) when x is not in that interval, extending the function by using its oddness, and then its parity. So at this point, they are saying, let's pick a value of x that is in [-4,0], and see if we can find f(x).

I think you may be making a mistake similar to the one Stephen appears to have made, misreading the problem. It doesn't say that x can only be in [0,4]; then they couldn't ask for f(198) at all, could they?

It tells you how f is defined for some values of x, and gives you additional information about how the function behaves for other values (parity and period), which you can use to extend the function to other values.

It will end up looking something like this:

View attachment 33138​

The red part is what they gave you, from 0 to 4. The green part is what you get when you apply oddness to that piece. The purple part is what you get when you use the period to extend to all real numbers. In the end, x can be any real number.

I still don't understand how f(x) = -f(x). It makes no mathematical sense, unless you use different domains for x for the left and the right hand side functions f, which is confusing. How can I mathematically come to the conclusion that f(x) = -f(x), without imagining a graph ? In general case, regardless of parity, f(x) can't be equal to -f(x) unless for function f(x) = 0, right ?

 

[JeffM]

Steven

The way the problem is worded (translated) is I believe intentionally obscure.

It says [imath]f(x) = x^4 - 14x^2[/imath] in the interval [0, 4].

It also says that f(x) is odd and periodic. Therefore, as I said in post 3, the formula for f(x) cannot possibly what is given for that one interval. The challenge is to find a general formula.

The problem is vaguely similar to, but much more sophisticated than, those asking about the continuity of a rational function that is not defined at some particular point but can be simplified to a rational function that is defined at that point.

 

[Dr.Peterson]

It seems that you are saying that one period of f is from [0,4].

Of course not. They tell us the period is 8. And they first extend it to [-4,4], which is one period.

On the interval [0,4] f, as far as I can see, is an even function.

What does that even mean?

I still don't understand how f(x) = -f(x).

Who said that?

I don't have time for a full answer; I'm on the way out the door!

 

[JeffM]

I still don't understand how f(x) = -f(x). It makes no mathematical sense, unless you use different domains for x for the left and the right hand side functions f, which is confusing. How can I mathematically come to the conclusion that f(x) = -f(x), without imagining a graph ? In general case, regardless of parity, f(x) can't be equal to -f(x) unless for function f(x) = 0, right ?

Unless f(x) = 0, no one says that f(x) = - f(x).

[math]f(x) = f(-x) \text { for } x \in [-a, a] \iff f(x) \text { is an even function in } [-a, \ a].\\ \text {Example: } f(x) = x^2.[/math]
Do you agree that [imath](-2)^2 = 2^2[/imath]?

[math]f(x) = - f(-x) \text { for } x \in [-a, a] \iff f(x) \text { is an odd function in } [-a, \ a].\\ \text {Example: } f(x) = x^3[/math]
Do you agree that [imath](-2)^3 = - 2^3[/imath]?

 

[Dr.Peterson]

The way the problem is worded (translated) is I believe intentionally obscure.

I wouldn't attribute intentionality to it without knowing more; but it is definitely not written for (or translated by) people who are not comfortable with these ideas!

For someone familiar with, say, Fourier series, it would be second nature.

On the interval [0,4] f, as far as I can see, is an even function. I think that you are saying that it is odd(??) How can that be??

You can't call a function on [0,4] even, because if f(x) is defined, then f(-x) is not defined!

What's happening here is that a function that incidentally happens to be even is being restricted to [0,4], and then extended to make an odd function (by reflection in the origin). This is a new function, not the original function!

Then this function on [-4,4] is being extended further as a periodic function (by repeated translation). And this function is what we are calling f!

I still don't understand how f(x) = -f(x). It makes no mathematical sense, unless you use different domains for x for the left and the right hand side functions f, which is confusing. How can I mathematically come to the conclusion that f(x) = -f(x), without imagining a graph ? In general case, regardless of parity, f(x) can't be equal to -f(x) unless for function f(x) = 0, right ?

Since no one said f(x) = -f(x), you'll have to explain where you see this stated in either the solution you were given, or what one of us has said. There is something that you are misunderstanding.

So that we're all on the same page, I got Google to translate the images, evidently from Serbian. Here is what it says, with a little tweaking by me:

22. If the function f is odd with period 8 and if f(x) = x^4 - 16x^2 for x ∈ [0,4], then f(198) is equal to: (A) 48 (B) -48 (C) 128 (D) -128 (E) 0​

​

22. Solution. Using the definition of the odd function f(-x) = -f(x) for each x ∈ D(f) we have that f(x) = -f(x). Thus, if x ∈ [-4,0] it follows that -x ∈ [0, 4], so​

f(x) = -[(-x)^4 - 16(-x)^2] = -x^4 + 16x^2.​

Therefore, the function f on [-4,4] is defined by​

f(x) = {x^4 + 16x^2, x ∈ [0, 4]​

.............{ -x^4 + 16x^2, x ∈ [-4,0].............(1)​

Since the function f is periodic with period 8 then​

f(x + k.8) = f(x), for x ∈ [-4, 4] and k ∈ Z..............(2)​

Since 198 = -2 + 8. 25 it is​

f (198) = f(-2 + 25.8) = f(-2) (due to relation (2))​

............. = -(-2)^4 +16(-2)^2 (due to (1))​

............. = 48.​

Note . It can be proved that if f is a periodic function with period T, then f(x + kT) = f(x), for every x ∈ D(f) and every k ∈ Z (see relation (2)). We have already used this equality for trigonometric functions, for example sin (x + k. 2π) = sin x, for every x ∈ R and every k∈ Z.​

I see that it is indeed their error, writing f(x) = -f(x) rather than f(x) = -f(-x) as I think they intended. I thought we had cleared this up, and you said you had copied incorrectly, but in either case knew that it was wrong. (I hadn't looked closely at the image of the solution until I worked through the translation.)

After making this correction, do you now understand everything? As I said at the start, "I think you have other issues, but I can't be sure until we clear this up."

 

[Dr.Peterson]

Looking back at the book's solution, I realized there is yet another typo in it (which I initially thought must be my own typo in the translation): The problem said

​

but in the solution they wrote

​

They never used this part, but it's still wrong. It's clear that they don't have a good proofreader!

@Ognjen: Since we've seen a couple typos and poor explanations, presumably in this same book, in other threads from you, we should keep in mind that it can't be trusted.

Whenever something doesn't make sense to you, don't assume it's your fault! Suppose that you are right, and continue reading with that assumption until it doesn't work. And when you ask for help, be sure to show all of what it says and ask questions about it, rather than about your own thinking, so we can catch those errors first, and then help you understand it.