# Checking if $f_1, f_2, f_3 Belong to$S_{X,3}$• MHB • mathmari In summary, the function f is continuous at every point in the interval$[-1,1]$, but is not of degree$3$. mathmari Gold Member MHB Hey! Let$S_{X,3}$be the vector space of the cubic splines functions on$[-1, 1]$with the points \begin{equation*}X=\left \{x_0=-1, \ x_1=-\frac{1}{2},\ x_2=0,\ x_3=\frac{1}{2}, \ x_4=1\right \}\end{equation*} I want to check if the following function are in$S_{X,3}$. 1.$f_1(x):=|x|^3$2.$f_2(x)=\left (x-\frac{1}{3}\right )_+^3$3.$f_3(x)=-x+x^3+3x^5$4.$f_4(x)=\sum_{n=0}^3a_nx^n$,$a_n\in \mathbb{R}, n=0, \ldots , 3$We have to check at each case if the function are of degree at most$3$and are$C^2$, or not? (Wondering) I have done the following: 1.$f_1(x):=|x|^3=|x|^3=\begin{cases}
x^3 \ \ \ ,& x\geq 0\\
-x^3 \ ,& x<0
\end{cases}$This function is continuous at every point, i.e. at$[-1, 0), (0, 1]$and at$x=0$. Then we have to check if the derivative id continuous. How can we calculate the derivative? (Wondering) 2.$f_2(x)=\left (x-\frac{1}{3}\right )_+^3$What exactly does the$+$mean? (Wondering) 3.$f_3(x)=-x+x^3+3x^5$This function is not in$S_{X,3}$, since it is of order$5$instead of at most$3$. 4.$f_4(x)=\sum_{n=0}^3a_nx^n$,$a_n\in \mathbb{R}, n=0, \ldots , 3$This function is$C^2$and of degree$3$. From that it follows that$f_4\in S_{X,3}$, right? (Wondering) mathmari said: [*]$f_1(x):=|x|^3=|x|^3=\begin{cases}
x^3 \ \ \ ,& x\geq 0\\
-x^3 \ ,& x<0

## 1. How do you define $S_{X,3}$?

$S_{X,3}$ is the set of all possible permutations of 3 elements from the set X. This means that each element in the set $S_{X,3}$ is a unique arrangement of 3 elements from the set X.

## 2. What is the purpose of checking if $f_1, f_2, f_3$ belong to $S_{X,3}$?

The purpose of this check is to determine if the given functions $f_1, f_2, f_3$ can be represented as permutations of elements from the set X. This is important in various mathematical and scientific applications, such as in group theory and combinatorics.

## 3. How can you check if $f_1, f_2, f_3$ belong to $S_{X,3}$?

To check if a function belongs to $S_{X,3}$, you can first list out all possible permutations of 3 elements from the set X. Then, you can compare the given functions $f_1, f_2, f_3$ to these permutations to see if they match. If there is a match, then the functions belong to $S_{X,3}$.

## 4. What are some examples of functions that belong to $S_{X,3}$?

Some examples of functions that belong to $S_{X,3}$ are: $f(x)=x^2$, $g(x)=\sin(x)$, and $h(x)=\frac{1}{x}$. These functions can be represented as permutations of elements from the set of real numbers, which is denoted as $S_{\mathbb{R},3}$.

## 5. Can a function belong to $S_{X,3}$ if X has less than 3 elements?

No, a function cannot belong to $S_{X,3}$ if X has less than 3 elements. This is because $S_{X,3}$ is specifically defined as the set of permutations of 3 elements from the set X. If X has less than 3 elements, there are not enough elements to form a permutation of 3 elements, and thus the function cannot belong to $S_{X,3}$.

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