Functions
Definition
A function is a rule relating to 2 sets1 in such a way that it assigns only 1 element of one set1 to an element of another set1
Programmatic Intuition
It can be looked at as a black box which takes one input, does something to it and produces an output.
Notation
Since function is a relation between 2 variables, we can write it as:
Where \(x\) is called the independant variable because it is our input of choice and \(y\) is called dependant variable since it depends of \(x\).
Domain and Range
Domain
The \(x\) comes from a set1 that is Set X called its domain.
Types of Domains
Natural Domain
This domain comes out as a result of the formula of the function itself.
Example
Here, the function becomes undefined due to division by zero at \(x = 2\).
Hence, the domain is \(\mathbb{R} - \{2\}\) or \((- \infty, 2) \cup (2, \infty)\).
Restricted Domain
Sometimes, the domain of a function can be altered by doing operations from algebra on the function to simplify it.
Example
We can simplify this and get
Notice how the domain of \(f(x)\) excludes 2 but in case of \(h(x)\) the function is actually defined.
Therefore, we have to write it as
Range
The \(y\) comes from a set1 that is Set Y called its range.
One of the techniques in finding range is to perform algebraic operations
Example
If we convert this equation to solve for \(x\), we get:
Clearly, we can see that \(y \ne 1\).
Therefore, the range is:
Piecewise Function
The graph of this function looks something like this.
Even And Odd
Even
A function is called even function if
Intuitively, this means that the function is mirrored by only \(y\) axis.
Odd
A function is called odd function if
Intuitively, this means that the function is mirrored by \(y\) axis and then mirrored copy is inverted by \(x\) axis.