# Can someone explain the domain and range (in functions) to me, and how to find them?

## I do not understand it. Like, at all.

Feb 15, 2018

Domain: all the values that can be used as INPUT in that function.
Range: the interval between the minimum and the maximum OUTPUTS obtained with that function.

#### Explanation:

Okay, it's really not complicated.
First, you need to understand what a function is.
Think of it like a vending machine.

You put coins in the vending machine, and it spits out the merchandise (drinks, snacks, whatever).
A function is like a vending machine.
Instead of coins, you put numbers in it.
And instead of drinks/snacks, you get a different number that the one you put in the first place.

The vending machine that is in my neighborhood only takes coins (say in US dollars) down to 5 cents, and up to 25 cents.
That is, it doesn't allow me to put in 50 cents coins (or any bills for that matter), nor does it allow me to put 1 cent coins.

Just like the vending machine, a function can have a limit to what numbers you can put in it.
This is called the "domain" of the function.
Let's say we have a function f with variable x, of the form:
$f \left(x\right) = x$
If they ask you "What is the domain of this function?", it really means "What are (all) the x you can put in and still get something out?".
In this case, the domain is ALL real numbers.
No matter what you put for x, you will get x.
So that's your domain. It goes from $- \infty$ to $+ \infty$.

Let's now say we have another function g with variable k, of the form:
$g \left(k\right) = \frac{1}{k}$
What is the domain in this case?
That is, what are all the value k that I can use in this function?
If you find yourself stuck, ask yourself the reverse question: what are the values of k where the function DOES NOT work?
In this case, you know that $\frac{1}{0}$ is not possible (division by zero is prohibited). Everything besides zero is fine, so, the domain of this function is ALL real numbers EXCEPT zero.

Now for the range.
The "range" is the interval from minimum value to the maximum value that the function spits you out!
Let's go back to our $f \left(x\right) = x$ function.
In this case, you get back whatever you put in, so the range in this case is the same as the domain. That is, it goes from $- \infty$ to $+ \infty$.
Let's try this other function $h \left(x\right) = \sin \left(x\right)$.
The sine function is known to oscillate with a maximum value of 1, and a minimum value of -1.
What is the domain? ALL real numbers.
What is the range? It is all real numbers in [-1; 1].

So, in summary, if you have a function f(x),
the domain of a function is : the interval of x you can INPUT in that function.
the range of a function is : the interval between the minimum and the maximum OUTPUTS obtained with that function

I hope these help.