Domain of a function
To understand what the domain of a function is, it is important to understand what an ordered pair is.
An ordered pair is a pair of numbers inside parentheses such as (5, 6).
Generally speaking you can write (x , y)
x is called x-coordinate and y is called y-coordinate
If you have more than one ordered pair, you name this situation set of ordered pairs or relation
Basically, the domain of a function are the first coordinates (x-coordinates) of a set of ordered pairs or relation.
For example, take a look at the following relation or set of ordered pairs.
( 1, 2), ( 2, 4), (3, 6), ( 4, 8), ( 5,10), (6, 12), (7,14)
The domain is 1, 2, 3, 4, 5, 6, 7. We will not focus on the range too much here. This lesson is about the domain of a function. However, the range are the second coordinates or 2, 4, 6, 8, 10, 12, 14
Let's say you have a business (selling books) and your business follows the following model:
Sell 3 books, make 12 dollars. (3, 12)
Sell 4 books, make 16 dollars. (4, 16)
Sell 5 books, make 20 dollars. (5, 20)
Sell 6 books, make 24 dollars. (6, 24)
The domain of your business is 3, 4, 5, and 6.
Pretend now that you can sell unlimited books. (3, 4, 5, 6, 7, ........).
Your domain in this case will be all whole numbers
You may then need a more convenient way to represent your business situation
A close observation at your business model and you will be able to see that the y-coordinate equals x-coordinate × 4
y = 4x
You can write (x, 4x). In this case, the domain is x and x represents all whole numbers or your entire domain for this situation.
In reality, it makes more sense for you to sell unlimited books.
Thus, when the domain is only 3, 4, 5, and 6, we call this type of domain restricted domain, since you restrict yourself only to a portion of your entire domain
In some cases, some value(s) must be excluded from your domain in order for things to make sense
Consider for instance all integers and their inverses as shown below with ordered pairs
...,(-4, 1/-4), (-3, 1/-3), (-2, 1/-2), (-1, 1/-1), (0, 1/0),(1, 1/1) (2, 1/2), (3, 1/3), (4, 1/4), ...
One of these these domain values will not make sense. Do you know which one?
It is 0. If the domain is 0, then 1/0 does not make sense since 1/0 is not defined or has no answer
Instead of writing all these ordered pairs, you could just write (x, 1/x) and say that the domain of definition is x such that x is not equal to 0
In general, the domain of definition of any rational expressions is any number except those that will make the denominator equal to 0
What is the domain of (6x + 7)/ x - 5
The denominator equals to 0 when x - 5 = 0 or x = 5
The domain for this rational expression is any number except 5
What is the domain of (-x + 5) / x
^{2} + 4
The denominator equals to 0 when x
^{2} + 4 = 0
However, x
^{2} + 4 is never equals to 0. Why? Because x
^{2} is always positive no matter what number you replace x with
4
^{2} = 16 and 16 is positive. 16 + 4 is still positive
(-5)
^{2} = 25 and 25 is positive. 25 + 4 is still positive
However, if you change the denominator to x
^{2} - 4, the denominator will be 0 for some numbers
x
^{2} - 4 = 0 when x = -2 and x = 2
2
^{2} - 4 = 2 × 2 - 4 = 4 - 4 = 0
(-2)
^{2} - 4 = 2 × 2 - 4 = 4 - 4 = 0
The domain will be in this case any number except 2 and -2
Consider now all integers and their square roots as shown below with ordered pairs
...,(-4, √-4), (-3, √-3), (-2, √-2), (-1, √-1), (0, √0),(1, √1) (2, √2), (3, √3), (4, √4), ...
Many of these domain values will not make sense. Do you know which ones?
They are -4, -3, -2, and -1. For any of these domain values, the square root does not exist. At least it does not exist for real numbers. It does exist for complex numbers, but this is a completely different story that we will not consider here
Our asumption here is that we are working with real numbers only to look for the domain of a function and the square root does not exist for real numbers that are negative!
Instead of writing all these ordered pairs, you could just write (x, √x) and say that the domain of definition is x such that x is bigger or equal to 0
What is the domain of √ (x - 5)?
When you deal with square roots, the number under the square root sign is called a radicand
√ (x - 5) is defined when the radicand x - 5 is bigger or equal to 0
x - 5 ≥ 0
x - 5 + 5 ≥ 0 + 5
x ≥ 5
The domain of definition is at least 5 or any number bigger or equal to 5
As you can see here, the domain of a function does not always make sense for some value(s). It is your job to find these values when you look for the domain of a function