Theory:

In this section, let us look at a few key terminologies relating to functions.

For that matter, let us consider the function $$f : X \rightarrow Y$$.

1. Domains and co-domains:

In the function $$f : X \rightarrow Y$$, the set $$X$$ is the domain, and the set $$Y$$ is the co-domain.

Domain $$=$$ Set $$X$$ $$=$$ $$\{x_1$$, $$x_2$$, $$x_3$$, $$x_4$$, $$x_5$$,$$...\}$$

Co-domain $$=$$ Set $$Y$$ $$=$$ $$\{y_1$$, $$y_2$$, $$y_3$$, $$y_4$$, $$y_5$$,$$...\}$$

2. Images and preimages:

If $$f(x) = y$$, the image of $$x$$ is '$$y$$' and the pre-image of $$y$$ is '$$x$$'.

From the figure given above, we can come to the following inferences.

For the image $$y_1$$, $$x_1$$ is its preimage.

For the image $$y_2$$, $$x_2$$ is its preimage.

For the image $$y_3$$, $$x_3$$ is its preimage.

For the image $$y_4$$, $$x_4$$ is its preimage.

For the image $$y_5$$, $$x_5$$ is its preimage.

3. Describing domain of a function:

Let $$f(x)$$ be $$\frac{1}{x^2 - 5x + 20}$$.

The function mentioned above holds for all real numbers except for $$4$$ and $$5$$.

In such cases, we can write $$f(x)$$ as $$\frac{1}{x^2 - 5x + 20}$$, where $$x \in R - \{4, 5\}$$.

4. Conditions to be a function:

$$f : X \rightarrow Y$$ can be called a function, only if:

• Every preimage of $$f$$ has an image.
• Each of the images is unique.

In figure 1, each of the preimages has unique images. Hence, figure 1 depicts a function.

Similarly, figure 2 is also a function.

But in figure 3, the preimage $$x_3$$ has the images $$y_2$$ and $$y_3$$. Since a preimage can only have one unique image, figure 3 does not represent a function. Also, $$x_2$$ does not have an image.

5. Range:

The set of images of $$f$$ is the range of that function.

From the image given above, Range $$=$$ $$\{y_1$$, $$y_2$$, $$y_3$$, $$y_4$$, $$y_5\}$$

Important!
Let $$n(K) = t$$, and $$n(L) = s$$.

Then the total number of functions between $$K$$ and $$L$$ is $$s^t$$.

For $$f : X \rightarrow Y$$, $$n(X)$$ $$=$$ $$2$$ and $$n(Y)$$ $$=$$ $$3$$.

The total number of elements in $$f$$ $$=$$ $$n(Y)^{n(X)}$$ $$=$$ $$3^2$$