### 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 \(B\).

Example:

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.

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.**\(f(x) = \frac{2}{x}\)