Theory:

Among the numerous functions available, there are a few unique cases of functions.
 
Let us look at a few of the special cases of functions.
 
1. Constant functions
 
2. Identity functions
 
3. Real valued function
 
Let us now look at each of them in detail.
1. Constant functions
A function \(A \rightarrow B\) is said to be a constant function if all the elements in \(A\) have the same image in \(B\).
Let us look at the example given below for a better understanding.
Example:
 
11.svg
 
In the example given above, all the preimages \(x_1\), \(x_2\), \(x_3\), \(x_4\) and \(x_5\) have the same image \(16\).
 
The given constant function can also be represented as \(f(x) = 16\).
2. Identity function
A function \(A \rightarrow B\) is said to be an identity function if the elements of \(A\) is equal to its image in \(B\).
That is, \(\text{Elements in }A = \text{Image in }B\).
Example:
Let us look at the example given below for a better understanding.
 
12.svg
 
Consider the image given above.
 
\(\text{Preimage in }A = x_1\)
 
\(\text{Its image in }B = x_1\)
 
Therefore, for every element in \(A\), there is an equal image in \(B\).
 
Also, the given arrow diagram can be represented as \(f(x) = x\).
3. Real valued function
A function \(f : A \rightarrow B\) is said to be a real-valued function if all the elements in the range belong to the set of real numbers, \(R\).
That is, \(f(A) \subseteq R\).
Example:
Some of the few examples of real valued functions are:
 
1. \(f(x) = 3x - 8\)
 
2. \(f(x) = x^3\)
 
3. \(f(x) = \frac{2}{x}\)