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The domain of a function \(y=f(X)\) is all possible \(x\) values it can take as inputs
The range is the set of all possible \(y\) values that it can output.
The domain and range are in the form of a group of numbers known as a set, and there are a few ways of representing this set:
1. Set Notation: Set notation is literally a list of numbers inside curly brackets like so \(\{x, y, z, ...\}\)
Note that this is not usable for continuous functions since they have infinite points. This is only usable on functions defind as a set of points
2. Set notation, but different: When you have a continuous function, the domain and range are instead both something of this form:
\( \{x\in \mathbb{R} | x \gt 3\} \)
The \( \in\) means "in", so \(x\) is in Real Numbers, and the "|" separates the second part, which is the rule. Basically it means, all real number values of \(x\), as long as they are greater than three.
3. Interval notation: This is the most convienient method of representing a set. Essentially, it is just \( (x,y)\) or \([x, y]\). It means all reals between \(x\) and \(y\), including any endpoint with a \([\) or an \(]\) and excluding any endpoint with a \((\) or an \()\). Note that for \(\infty\), you always use circular brackets, nevdr squeare
In finding domain and range, it is generally good to think of the type of functions that you are dealing with. For example, a linear function will all real numbers as both domain and range, while a square root function will have all nonnegative reals as a domain. The functions may be modified, but you can account for that by tweaking the domain and ranges. You will see how in the next example.
Find the range of \(f(x) = 3\sqrt{x-5} -2\) and give it in set and interval notation.
First, we find the range of \(f(x)\). Normally a square root function has a range of all nonnegative reals, but this function has been shifted down by two, so when the function would give zero, it actually gives minus two, so the range starts two further down.
Interval Notation:
$$(-2, \infty)$$
Set Notation:
$$\{x\in \mathbb{R} | x \gt 2\}$$
In the next example, we will do the same thing for a rational function.
Also, multiple intervals can be joined using the \(\cup\), which is called a Union. This will be demonstrated in the next example.
Find the domain \(f(x) = \frac{1}{x-4}\) in set and interval notation
In math, you cannot divide by zero, so the functions domain will not include any values of \(x\) that that make the denominator equal to zero. In this case, the only value of \(x\) that does this is \(x=4\). This makes the set notation quite simple. $$\{x\in \mathbb{R} | x \gt 2\}$$ The interval notation is a bit different. Because it is all real numbers with a gap in the middle, it can only be represented as the combination of two sets, \((-\infty, 4)\) and \((4, \infty)\). Therefor, it will be written using the \(\cup\) symbol as $$(-\infty, 4)\cup(4, \infty)$$
In the final example, we will find the domain and range of a function given by a set of points.
Find the domain and range of the function \(f(x)\) defined by:
(1, 1), (2, 5), (3,2), and (4, 1)
Since the domain is all values for which a function is defined, the domain is just every \(x\) for which there is a point. Since this is a set of points and not a continuous function, we will use the first kind of set notation: $$\{1, 2, 3, 40\}$$ The range will be slightly different. Although both \(x = 4\), and \(x = 1\) give \(y =1\), we don't put it down twice. Instead the set is just: $$\{1, 5, 2\}$$
Conlcusion:
You learned how to find the domain and range of functions, and three ways of representing it.
Next: One to One functions