### 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:

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