# Thread: Describing the domain and range.

1. ## Describing the domain and range.

One question on my take-home exam is this:

For each of the following types of functions, give an example in algebraic form f(x)= (insert function), then sketch the graph and describe the domain and range.

I've got an example of each function asked of me (linear, constant, quadratic, radical, exponential, logarithmic) and I've also done the graph of each. I know what domain and range is but "describe" it? I emailed my prof. asking her what exactly does she mean by "describe it" and she said; "Interval notation for the domain and range is one form of a good description. (Set builder notation and verbal descriptions can also work.)

I must admit that putting domain and range in interval notation is not my strong point. So that is what I'd like help with.

If it helps, here are the examples I used:

Linear f(x) = 2x+4
Constant f(x) = 2
Radical f(x) = square-root(3-x) (I'm sorry I don't know the command for square root).
Exponential f(x) = 2^x
Logarithmic f(x) = log 2^x (again sorry for not knowing the command to display the 2 lower than the log).

If anyone is able to help me understand how to find the interval notation from these examples that would be super-amazing-great. My favorite math site, Purple Math, isn't as helpful as usual with this subject. My 75 dollar online textbook requires me to basically read a book just to figure out one thing.

Thanks in advance and sorry again for my MathHelpForum newbieness.

Edit: If you'd prefer to talk about set builder notation and/or verbal descriptions that would be helpful as well, as my prof. said those were acceptable methods too. I chose interval notation because I think that's her favorite.

2. Interval notation, as you may know, is just listing the intervals over which a function is defined. For example, if you see

Range: $\displaystyle [-1, 3)$

that means that the range of the function is all the values from -1 to 3, including -1 but not including 3. (The bracket means including; the parenthesis means not including.) A lot of functions have domains or ranges which extend all the way from negative infinity to infinity. This is written

Domain: $\displaystyle (-\infty , \infty)$

which means that x can take on any value, all the way from negative infinity (that means any negative number) up to positive infinity (zero and any positive number). Notice that we use parentheses when dealing with infinity, which means that x can take on any value up to, but not including, infinity. That's because infinity is not a real number; it's just a shorthand for an idea - no number can actually "equal" infinity.

Set builder notation is just a little different. Instead of dealing with intervals, it usually deals with inequalities or other conditions. So to write "the domain of f(x) is all numbers such that x is not equal to zero and x is not equal to 4" in set builder notation is

$\displaystyle x\mid x\not =0, x\not = 4$

A breakdown of this notation is

$\displaystyle x$ means "all x"
$\displaystyle \mid$ means "such that"
$\displaystyle x$ means "x"
$\displaystyle \not =$means "is not equal to"
$\displaystyle 0$ means "0"
$\displaystyle ,$ means "and"
$\displaystyle x$ means "x"
$\displaystyle \not =$ means "is not equal to"
$\displaystyle 4$ means "4"

A verbal description, which is probably your best bet if you don't have a lot of time to learn this stuff in one night, is simply describing the domain or range in words like I did above. However, this can get inelegant with more complicated functions. Example:

"The domain of f(x) is all numbers below or equal to three, and all numbers above four."

Here's a breakdown of the three methods, using the domain "all real numbers" - ie, the case in which the function can take on any value for x.

Interval notation: $\displaystyle (-\infty , \infty)$
Set builder notation: $\displaystyle x\mid x\in \mathbb{R}$
Verbal description: "All real numbers"