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,Originally Posted by batman123
$\displaystyle f(x)=-3x^2$ 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,
$\displaystyle y=\sqrt{\frac{1}{x}}$
Got a little "oopsie" here. This should be "notice that $\displaystyle -3x^2$ is never a positive number thus, $\displaystyle y \in (- \infty,0]$.Originally Posted by ThePerfectHacker
-Dan
(Hey, usually I'm the guy getting the set notation wrong! )
Hello,Originally Posted by CaptainBlack
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