A function is a relation between two sets of variables such that one variable depends on another variable. We can represent different types of functions in different ways. Usually, functions are represented using formulas or graphs. There are four ways for the representation of a function as given below:

• Algebraically
• Numerically
• Visually
• Verbally

Each one of them has some advantages and disadvantages. Let us look at them once and try to understand them.

Representation of a Function – Algebraic

The algebraic representation of a function refers to the expression of a function using an equation or mathematical model. As we know, the process of interpreting a real-world problem in a convenient function is called mathematical modelling. The resulting function and a representation of all input and output variables, including proper units of measure are referred to as a mathematical model. Input/output diagrams often represent mathematical models. 

It is one of the usual representations of functions. Here, functions are explicitly represented using formulas, and the functions are generally denoted by lower case alphabet letters. Let us take the cube function.

Figure 1: Block diagram depicting a cube function

The standard letter to represent function is f. However, it can be represented by any variable. To denote the function f algebraically, i.e., using the formula, we write:

\( f:x \)→ \( x^3\)

where x is the variable indicating the input. It can be represented by any variable.

\( x^3 \) is the formula of function

f is the name of the function

Though one of the easy and understandable ways of representing a function, it is not always easy to get the function’s formula. For those cases, we use other methods of representation.

Representation of a Function- Visual

This is basically the graphical representation of functions. This way of representation is very easy to understand. The input values are marked along the x-axis. For any input value, the corresponding output value is the vertical displacement from the x-axis. For e.g. at x = a, the output is equal to f(a).

Figure 2: Graph of a function

The graph shows the properties of the functions. For e.g. from figure 2, we can directly tell:

• where the graph is increasing or decreasing
• where the rate of change is more and where it is less
• where are the extreme values

Thus, graphs are very beneficial for studying the behaviour of the function. One drawback is that we can’t always get the exact values of all the outputs from the graph.

Learn more about functions and their graphs here.

Representation of a Function- Numerical

This is basically the tabular way of representing a function. The table contains two columns; one with the dependent variable and the other with the independent variable. To show with an example, let us take a function f and independent variable as x. The table is given as:

Table 1: Table representing a function f(x) = 2x

Though we have the exact value of the outputs, we can only have a finite number of such outputs. The analysis of the function and study of its behaviour hence becomes difficult.

Representation of a Function- Verbal

In this way of representing functions, we use words. For e.g.

• For the input x, the function gives the largest integer smaller than or equal to x i.e. floor function (see fig. 3).
• For the input x, the function gives the value equal to x i.e. identity function (see fig. 4).

Figure 3: Floor function

Figure 4: Identity function

Each of the representations has its pros and cons. According to the information required, appropriate representation should be adopted. To learn more about functions, visit BYJU’S and fall in love with learning!