Set Notation for Domain and Range

• Mar 5th 2006, 12:30 PM
batman123
Set Notation
Using some form of set notation, indicate the domain and range of each function defined below:

f(x)=-3x^2 Domain: Range

f(x)= 5x-7 Domain: Range:

f(x)= -2
________
7 - x

Domain: Range:
• Mar 5th 2006, 01:55 PM
ThePerfectHacker
Quote:

Originally Posted by batman123
Using some form of set notation, indicate the domain and range of each function defined below:

f(x)=-3x^2 Domain: Range

f(x)= 5x-7 Domain: Range:

f(x)= -2
________
7 - x

Domain: Range:

Notice that the function,
$\displaystyle f(x)=-3x^2$ has a meaning for any $\displaystyle x$. Now you want it in set notation, one way is to write $\displaystyle x\in \mathbb{R}$ implying that $\displaystyle x$ is any REAL number. Another way is to assign the interval $\displaystyle x\in (-\infty,\infty)$. Meaning $\displaystyle x$ could be as small as you wish and as large as you wish. Now the range is the $\displaystyle y$ value, notice that $\displaystyle -3x^2$ is always a negative number thus, $\displaystyle y \in [0,\infty)$. Notice the square bracket indicating that $\displaystyle y$ can be equal to 0 which is true.

The function $\displaystyle f(x)=5x-7$ is defined for any $\displaystyle x$ thus we write $\displaystyle x\in (-\infty,\infty)$ (let me just remind you then infinity is not a number even I use it does not mean that it is). If you graph this line you will see the the $\displaystyle y$ value is anything thus we write $\displaystyle y\in (-\infty,\infty)$.

The function $\displaystyle f(x)=\frac{-2}{7-x}$ is not defined for all $\displaystyle x$ it is undefined for $\displaystyle x=0$ thus, $\displaystyle x$ cannot be 0, thus we write (this might be tricky) $\displaystyle x\in (-\infty,0)\cup (0,\infty)$- the circle bracket indicate $\displaystyle x\not =0$ which we are trying to state. If you graph this function, you will see the $\displaystyle y$ always has a value except for $\displaystyle y=0$ (because the equation $\displaystyle 0=\frac{-2}{7-x}$ has not solution). But besides for that $\displaystyle y$ always has a value thus we write, $\displaystyle y\in (-\infty,0)\cup(0,\infty)$.

Let me give you a more challenging one,
$\displaystyle y=\sqrt{\frac{1}{x}}$
• Mar 5th 2006, 02:52 PM
topsquark
Quote:

Originally Posted by ThePerfectHacker
notice that $\displaystyle -3x^2$ is always a negative number thus, $\displaystyle y \in [0,\infty)$

Got a little "oopsie" here. This should be "notice that $\displaystyle -3x^2$ is never a positive number thus, $\displaystyle y \in (- \infty,0]$.

-Dan

(Hey, usually I'm the guy getting the set notation wrong! :D)
• Mar 5th 2006, 03:05 PM
batman123
Thanks!
Thanks Guys
• Mar 5th 2006, 03:14 PM
ThePerfectHacker
Quote:

Originally Posted by topsquark
Got a little "oopsie" here. This should be "notice that $\displaystyle -3x^2$ is never a positive number thus, $\displaystyle y \in (- \infty,0]$.

-Dan

(Hey, usually I'm the guy getting the set notation wrong! :D)

I always thought I was Perfect, now you showed that even I am able to make mistakes.
• Mar 5th 2006, 08:45 PM
earboth
Quote:

Originally Posted by ThePerfectHacker
...

The function $\displaystyle f(x)=\frac{-2}{7-x}$ is not defined for all $\displaystyle x$ it is undefined for $\displaystyle x=0$ thus, ...

Hello,

... it is undefined for $\displaystyle 7-x=0$ because the division by zero is not allowed. Thus the domain is $\displaystyle \Re \setminus \{ 7 \}$

Greetings

EB
• Mar 5th 2006, 11:10 PM
CaptainBlack
Quote:

Originally Posted by earboth
Hello,

... it is undefined for $\displaystyle 7-x=0$ because the division by zero is not allowed. Thus the domain is $\displaystyle \Re \setminus \{ 7 \}$

Greetings

EB

The usual symbol for the reals is $\displaystyle \mathbb{R}$ obtained with
the TeX string "\mathbb{R}".

RonL
• Mar 6th 2006, 12:14 AM
earboth
Quote:

Originally Posted by CaptainBlack
The usual symbol for the reals is $\displaystyle \mathbb{R}$ obtained with
the TeX string "\mathbb{R}".

RonL

Hello,

Thanks for your hint. I've taken the symbol which I found in the "Latex-manual" of the forum. Maybe it would be nice to add those symbols to this manual. (Are there any \matbb{N}, or \mathbb{Z}, or ... too?)

Once again: Thanks.

Greetings

EB