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.