Representation of a Function

Functions, as important they are, is their representation. Usually, functions are represented using formulas or graphs. But there are other ways in which functions can be represented. There are four ways for representation of a function:

  • Algebraically
  • Numerically
  • Visually
  • Verbally

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

Representation of a function- Algebraic

It is one of the usual representations of functions. In this, functions are explicitly represented using formulas. The functions are generally denoted by lower case alphabet letters. For e.g. let us take the cube function.

Figure 1: Block diagram depicting 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 formula, we write:

\( f:x \)→ \( x^3\)

where x is the variable denoting 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 formula of the function. For those cases, we use other methods of representation.

Representation of a function- Visual

This is basically the graphical representation of functions. This 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 extrema

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

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 other with the independent variable. To show with an example, let us take a function f and independent variable as x . So, the independent variable will be equal to f(x) . The table is given as:

Table 1: Table representing a function

x F(x)
-1 -1
– 0.5 -1
0 0
0.5 0
1 1
1.5 1

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 behavior 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