# Valentina Franceschi

I am Postdoc researcher at Inria Paris. My research field is mainly sub-Riemannian geometry. Among my interests: photography, swimming and science communication.

Location Paris, France

## Activity

• Dear Miguel,

Try to stop the video as you need some more time to read the slides :)

Hope this helps!

Valentina

• Dear Sam,

It is normal! Learning how to construct clear arguments that lead to the solution is the hard part of the work :)

Bests

Valentina

• Dear Ged,

As you suggest at the end of your comment, the first sentence in your comment is true if you add the word "sometimes", otherwise it is false.

Bests

Valentina

• Hi Nicola,

Ambient set means the "surrounding set". Namely, in this case, as the range is included in the codomain, you set that the codomain is the ambient space for the range. It is the (possibily bigger) set where the range lives.

Bests

Valentina

• Dear Ged,

In exercise one the range is ]-Infinity,-1[U]0,Infinity[.

In exercise 2 it is R except 1. You can prove it by looking for solutions of g(x)=y. As explained in the pdf you will see that it admits solutions only for y different from 1.

Bests

Valentina

• Dear Ged,

Just corrected. Thank you very much

Valentina

• Dear Ged,

Your argument is correct, hence the sentence is false. Indeed, as you can choose √3 and -√3 whose sum is rational, it is not true that for any choice of irrational numbers the sum is irrational.

Best

Valentina

• Dear Ged,

1) It is an element of B (If x∈B means that we are assuming x to be an element of B).
2) Yes, you can add a ALSO for a better comprehension
3) These two sets are equivalent
4) Correct! In fact, it is useful to write this expression because you want to use the assumptions and prove that B=C. The idea is then to...

• Dear Graham,

Regarding your first question: yes, you are right!

Regarding the special role of the Euler constant e in mathematics, besides the compound interest, there are several relevant properties regarding, for instance, the complex exponential and its relations to trigonometric functions, Gaussian functions used in probability, and the slope of...

• Dear Fabien,

Very good! Indeed it is true that cos is an even function, that is cos(x)=cos(-x) for every real x.

Valentina

• Dear Graham,

That's correct.

Best

Valentina

• Hi everybody,

Great observation!

If you want to define sin and cos starting from a circle of a different radius you just have to divide the length of the x and y coordinates of your point on the circle by the radius itself. That's why you always consider radius 1.

Best

Valentina

• Dear Joshua,

It means that phi is a function with domain D and codomain E. Namely, phi(x)\in E for every x in D.

I hope this helps

Valentina

• Dear Clare,

Might it have been the letter "phi"? If not, could you please give us a more precise reference to the lecture?

Valentina

• Thank you for your suggestion, Cruz.

• Dear B M,

Several mathematicians define the set of natural numbers starting from 0, several others from 1. With your definition of natural numbers, your answer is correct.

Best

Valentina

• Dear Juli,

That's correct!

Anyway, I've corrected the text of the exercise.

Thank you!

Valentina

• Thank you Graham!
Hope to meet you again in some pre/calculus course soon!

• Thank you Leigh!

• Dear Graham,

I guess you are talking about Quiz 4.14.

Both corrections have been implemented.

Thank you for you suggestions!

Valentina

• @IvanaBabkova Dear Ivana,

There was a mistake!
The correct solution is now displayed.

Thank you!

Valentina

• Dear Ivana,

That's correct! Great!

Valentina

• Correct!

We will fix it!

Valentina

• Correct! We will fix it. Thank you!

Valentina

• Dear Taj,

This is 1/(25^2/3) by definition of negative power.
Then 25^2/3 is \sqrt{3}{25^2} by definition of third root of a number.

Hope this helps,

Valentina

• At the beginnings it might look hard to understand, but don't give up and you will see that it's a useful notation to describe things!

If your doubts come from the fact that infinity is not a real number, try to think in the following way. A number is something that you can approximate with some accuracy with a rational number. Namely a real number is close...

• Dear Ivana,

Your attempt is great! The computation of the rotation angle usually involves some linear algebra. Actually, I get a different result, but let me see your solution.
How did you compute the rotation?

Valentina

• Dear Graham,

I can see the same problem in the video.
I am not sure it will be possible to fix it, though.

Hope it is not too annoying.

Valentina

• Correct!

Actually infinity is not a number. [a,+infinity[ is just a notation to say that you include all greater and greater real numbers.

Valentina

• @GrahamGardiner Ok! It is just a different way of saying the same thing.

Thank you Graham

Valentina

• Dear Graham,

Correct!

Thank you!

Valentina

• Dear x Svargo and Mel

The notation [3,6[ means all the REAL numbers that are smaller than 6 and greater than or equal to 3. The notation {3,4,5} means only the INTEGER numbers. For instance, the number 3.5 is in the interval [3,6[, but not in the set {3,4,5} since it is not integer.

Hope this helps.

Valentina

• Dear Sling,

We just put it.

Valentina

• Dear Graham,

Just to be sure that we agree with definitions, I propose a different example.

Consider the function f:(0,1)->(0,1), f(x)=x^2. Its inverse is f^{-1}:(0,1)->(0,1),
f^{-1}(x)=sqrt{x}. You can find it by solving
y=f(x)=x^2 iff x=sqrt y=f^{-1}(y).

The graphs of the two functions are different.
Nonetheless, by definition y=f(x) iff...

• @GrahamGardiner Dear Graham,

I don't use geogebra, so I don't know if there is the possibility of specifying the domain definition. In this case, if you plot the fanction from the y variable to the x one, I am sure you will see exactly the inverso on its domain.

Let me know!

Valentina

• @timbloore Yes! It is the mathematical/logical formulation of an everyday concept (being greater than or equal to something)

Valentina

• @GrahamDivall Dear Graham, we will correct it as soon as possible.

Valentina

• Thank you Ted!

Just fixed it.

Valentina

• Thank you Graham!

• Dear Michael,

We are assuming a and b to be positive and integers, so that a^2 and b^2 are positive and integers. The equation a^2=2b^2 implies that a^2 is twice the number b^2. Namely a^2 is the integer number b^2 multiplied by 2.

Hope this helps.

Valentina

• Dear Paul,

First of all, sorry for the miswriting then --> than :)

Regarding your question: x>y does not include x=y, indeed.
In particular, x>=y does not imply x>y.
For instance: 2>=2, but it is not true that 2>2.

To say x>=y you have to check either one of the two propositions x=y or x>y.

Hope this helps

Valentina

• Dear Graham,

It is true that an injective function can be reversed ON ITS RANGE.
Namely, if f:E->F is injective and f(E)=G, then there exists the inverse f^{-1}:G->E.

If you want to invert the function on the whole codomain you also need to ask for a surjective function, in such a way that the range and the codomain correspond.

Hope this...

• Dear Graham,

I think you got the point!

As you say, "the point x,y occurs on the graph of the inverse function with the values of x and y exchanged" means that the couple (y,x) occurs on the graph of the inverse function, namely that x=f^{-1}(y). This is the proper way to say it because you define the graph of a function as the set of points where the...

• Correct!
Thank you Graham and Robin.

Valentina

• Correct! Thank you Ivana!

Valentina

• Dear all,

As @FrankPrice said, you say that a number x is "greater than or equal to" a number y if one of the two following cases occur: either x is "strictly greater than" y, or x is "equal to" y. For example: 3 is greater than or equal to 2 because it is strictly greater than 2, but also 2 is greater than or equal to 2 because it is equal to 2.

From...

• Dear Felix,

It is any number that you can write as a fraction a/b, where a is an integer number and b is a natural one (which is not 0). These numbers are not all the real numbers. For instance square root of 2 is not a rational number, namely you cannot write it as a fraction.

Valentina

• Dear Liz,

What Quiz are you talking about?

Valentina

• Dear all,

That's correct.
I just fixed the solution.

Valentina

• Thank you Ted,

Just fixed it.

Valentina

• Thank you Ivana,

We will fix this.

Valentina

• Dear Hanouf,

You can think that the rule [ (a^p/a^q)=a^p-q ] follows to from the following rule:
[ (a^m a^n)=a^m+n ] for m=p, n=-q. In fact, in this case a^-q=1/a^q.

Valentina

• Thank you Ted, corrected.

Valentina

• @TonySamson Dear Tony,

I confirm what Ivana and Graham told you. Logarithms must have base bigger then 0, not then 1. In particular, logarithms with base smaller then 1 have a different monotonicity with respect to the others.

Regarding the discussion. I will talk to the other Educators to see what are the possibilities!

Thank you all for your...

• @JuliSibi Done! Thanks for the suggestion.

Valentina

• Dear Graham,

Valentina

• Dear Graham,

No, there doesn't exists a terminology for many-to-one.

If the function is not one-to-one you simply say that it is not injective.

I would say that the motivation for not giving a precise name to many-to-one functions, in comparison with one-to-one functions, is that they cannot be inverted as functions from their domain to their range....

• Dear Graham,

Your reasoning to write the laws and to find the inverse laws is correct.

Regarding the domains:
From the graph you can see that the domain of f is [-1,3]. (Anyway, you can imagine -1 and 3 to be out of the domain as you wish. In this case you would have (-1,3) or [-1,3) or (-1,3]. It is just important that the continuation of your...

• Dear Graham,

1. Yes, correct.

2. I am not sure that I understood your question. What happens is that f(x)=y iff f^{-1}(y)=x.

3. It is correct when you write the explicit rule of the inverse. (In fact you could also call the variable a or whatever you like here).
It is not correct when doing computations to explicitly find the inverse. For instance,...

• Dear Graham,

It is just a misprint. The correct solution id (3,5].

Valentina

• Dear Graham,

We are not giving the definition of the fifth root of a negative number. We are just saying that if you take the fifth root of the positive number (4/9), you multiply it by minus one and then you take the fifth power of this product, you end up with the desired solution.

Valentina

• Dear Liz,

What you say is correct. Actually if a number is strictly less then two, of course it is also less then or equal then two. In general, if x < y then it is also true that x <= y because y <= y. Does it help the understanding?

Valentina

• Until the post will not be corrected, here is a link to the correct video : https://mediaspace.unipd.it/media/W4_SeveralVariables_1/1_n6wtf009

• Dear Graham,

As it is posed the question means: Does the equation admit an infinite number of solutions for any choice of a?", namely: Is it true that, given a real, there are infinite real numbers (x) that solve the equation?"

Valentina

• Dear Graham

It is true that a = 2 gives two different roots, but the answer is not correct because a=2 is not the only value for which two distinct solutions are obtained.

Valentina

• Dear Graham and Ivana,

What Ivana says is correct.
Maybe the part that is missing in the calculation is

(- 5th root of 4/9)^5
= ( (-1) x (5th root of 4/9))^5
= (-1)^5 x (5th root of 4/9)^5
= -1 x 4/9
= -4/9

Does it help the understanding?

Valentina

• Dear Liz,

Regarding Q3 there were indeed two correct answers, I just modified the quiz in order to have only one correct answer. Do you still see no correct answers?

Regarding Q5, you are right and I just corrected the explanation.

Thank you very much

Valentina

• Dear Johnnie,

Yes, here we are always dealing with real roots.

Valentina