2. Terminologies related to functions

Let us look at a few key terminologies relating to functions in this section.

 

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 draw 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:

 

$f : X \rightarrow Y$ is only a function if and only if the following conditions are met:

  

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.