Representation of Functions: A Guide for 10th Grade Mathematics

Representation of Functions - Mathematics

Representation of Functions

A function may be represented by

• (a) a set of ordered pairs
• (b) a table form
• (c) an arrow diagram
• (d) a graphical form

Let \(f: A \rightarrow B\) be a function

(a) Set of ordered pairs

The set \(f = \{(x, y) | y = f(x), x \in A\}\) of all ordered pairs represent a function.

(b) Table form

The values of \(x\) and the values of their respective images under \(f\) can be given in the form of a table.

(c) Arrow diagram

An arrow diagram indicates the elements of the domain of \(f\) and their respective images by means of arrows.

(d) Graph

The ordered pairs in the collection \(f = \{(x, y) | y = f(x), x \in A\}\) are plotted as points in the \(xy\)-plane. The graph of \(f\) is the totality of all such points.

Every function can be represented by a curve in a graph. But not every curve drawn in a graph will represent a function.

The following test will help us in determining whether a given curve is a function or not.

Vertical line test

“A curve drawn in a graph represents a function, if every vertical line intersects the curve in at most one point.”

Example 1.10

Using vertical line test, determine which of the following curves (Fig.1.18(a), 1.18(b), 1.18(c), 1.18(d)) represent a function?

Solution

The curves in Fig.1.18(a) and Fig.1.18(c) do not represent a function as the vertical lines meet the curves in two points \(P\) and \(Q\).

The curves in Fig.1.18(b) and Fig.1.18(d) represent a function as the vertical lines meet the curve in at most one point.

Notes

Any equation represented in a graph is usually called a ‘curve’.

Example 1.11

Let \(A = \{1, 2, 3, 4\}\) and \(B = \{2, 5, 8,11,14\}\) be two sets. Let \(f: A \rightarrow B\) be a function given by \(f(x) = 3x - 1\). Represent this function

1. by arrow diagram
2. in a table form
3. as a set of ordered pairs
4. in a graphical form

Solution

Given: \(A = \{1, 2, 3, 4\}\); \(B = \{2, 5, 8,11,14\}\); and \(f(x) = 3x - 1\).

\(f(1) = 3(1) - 1 = 3 - 1 = 2\)

\(f(2) = 3(2) - 1 = 6 - 1 = 5\)

\(f(3) = 3(3) - 1 = 9 - 1 = 8\)

\(f(4) = 3(4) - 1 = 12 - 1 = 11\)

(i) Arrow diagram

Let us represent the function \(f: A \rightarrow B\) by an arrow diagram (Fig.1.19).

Fig.1.19

(ii) Table form

The given function \(f\) can be represented in a tabular form as given below.

(iii) Set of ordered pairs

The function \(f\) can be represented as a set of ordered pairs as
\(f = \{(1,2),(2,5),(3,8),(4,11)\}\)

(iv) Graphical form

In the adjacent \(xy\)-plane the points (1,2), (2,5), (3,8), (4,11) are plotted (Fig.1.20).

Fig.1.20