# Thread: Consider the function f(x)=2-3x with domain R ( real numbers). Let A = [-1,3), B=

1. ## Consider the function f(x)=2-3x with domain R ( real numbers). Let A = [-1,3), B=

This is from a a question i posted a while back but never got a good response to.

"Consider the function f(x)=2-3x with domain R ( real numbers). Let A = [-1,3), B=(2,5]."

now another question is find f|A composite XB (big with with subscript B)

My teacher showed the graph for it. But i dont understand it.

Could someone please help in how this graphs comes to be?? Thanks.

2. ## Re: Consider the function f(x)=2-3x with domain R ( real numbers). Let A = [-1,3), B=

So, we need to find $\displaystyle f|_A\circ\chi_B$ where $\displaystyle f|_A$ is the restriction of $\displaystyle f$ to $\displaystyle A$ and $\displaystyle \chi_B$ is the characteristic function of $\displaystyle B$ defined by

$\displaystyle \chi_B(x)=\begin{cases}1&x\in(2,5]\\0&\mbox{otherwise}\end{cases}$

Do you understand this definition of $\displaystyle \chi_B(x)$? This is a so-called piecewise-defined function. Can you find $\displaystyle \chi_B(0)$, $\displaystyle \chi_B(2)$, $\displaystyle \chi_B(3)$, $\displaystyle \chi_B(5)$, $\displaystyle \chi_B(8)$? What are all possible values of $\displaystyle \chi_B(x)$?

3. ## Re: Consider the function f(x)=2-3x with domain R ( real numbers). Let A = [-1,3), B=

Yes i understand the definition. i understand the graph of XB. I dont understand what is meant by XB(0), XB(1).. and so forth though

4. ## Re: Consider the function f(x)=2-3x with domain R ( real numbers). Let A = [-1,3), B=

I am genuinely curious about what you don't understand. First note that when I write

$\displaystyle \chi_B(x)=\begin{cases}1&x\in(2,5]\\0&\mbox{otherwise}\end{cases}$

$\displaystyle \chi_B$ is being defined. The meaning of $\displaystyle \chi_B$ is not composed of the meanings of $\displaystyle \chi$ and $\displaystyle B$, as if you are supposed to know them already. No, one can replace $\displaystyle \chi_B$ by some symbol $\displaystyle f$, and this would be a definition of a new function $\displaystyle f(x)$.

Could you explain how it is possible to understand the definition of a function and the graph of a function but not understand how to apply the function to a specific value? Do you understand the general concept of a function? Could you be more specific about what you do and do not understand?

5. ## Re: Consider the function f(x)=2-3x with domain R ( real numbers). Let A = [-1,3), B=

I think its the whole concept of the characteristic function. it's only mentioned in a brief paragraph in our book. There really isn't any reasoning behind it. Our teacher didn't explain the purpose of what this is applied too, just decided give a question over it. So i don't know what to do when given this question. I hardly understand the notation. Thanks anyway guys .

6. ## Re: Consider the function f(x)=2-3x with domain R ( real numbers). Let A = [-1,3), B=

The composition $\displaystyle f|_A\circ\chi_B$ can be described by the following situation. Suppose you are a supervisor and you want to meet with your team. You decide do hold two meetings. You give each team member a paper slip with either 0 or 1 written on it. Then you tell everybody that if they get a slip with x written on it, they should come to the meeting at 2 - 3x pm. So, those with 1 come at 2 - 3(1) = -1 pm (let's agree that this is 11 am) and the rest come at 2 pm.

Let B be the group that received 1. The function that for each person returns 0 or 1 is called the characteristic function of the group B; it sets apart people in B by saying that each person in that group received 1. The composition $\displaystyle f|_A\circ\chi_B$ is analogous to a function that, given a person, returns the time of the meeting.

7. ## Re: Consider the function f(x)=2-3x with domain R ( real numbers). Let A = [-1,3), B=

Another minor note: function composition (unfortunately) works in reverse. So you take a real number and plug it into the characteristic function for B before plugging the result into f. Since the range of B is {0, 1} and this lies in the domain of f|A, the function is defined everywhere on R. If A were instead, say, (0, 5], instead of taking value -1 in the range (2, 5] and 2 elsewhere, it would take value -1 in the range (2, 5] and be undefined elsewhere.