“ THE GRAPHS AND THEIR SHAPES OF FUNCTION “

In mathematics, the graph of a function f is the collection of all ordered pairs (x, f(x)). If the function

input x is a scalar, the graph is a two-dimensional graph, and for a continuous function is a curve. If

the function input x is an ordered pair (x1, x2) of real numbers, the graph is the collection of

all ordered triples (x1, x2, f(x1, x2)), and for a continuous function is a surface (see three-dimensional

graph).

Informally, if x is a real number and f is a real function, graph may mean the graphical representation

of this collection, in the form of a line chart: a curve on a Cartesian plane, together with Cartesian

axes, etc. Graphing on a Cartesian plane is sometimes referred to as curve sketching. The graph of

a function on real numbers may be mapped directly to the graphic representation of the function. For

general functions, a graphic representation cannot necessarily be found and the formal definition of

the graph of a function suits the need of mathematical statements, e.g., the closed graph

theorem in functional analysis.

The concept of the graph of a function is generalized to the graph of a relation. Note that although a

function is always identified with its graph, they are not the same because it will happen that two

functions with different codomain could have the same graph. For example, the cubic polynomial

mentioned below is a surjection if its codomain is the real numbers but it is not if its codomain is

the complex field.

To test whether a graph of a curve is a function of x, use the vertical line test. To test whether a

graph of a curve is a function of y, use the horizontal line test. If the function has an inverse, the

graph of the inverse can be found by reflecting the graph of the original function over the line y = x.

In science, engineering, technology, finance, and other areas, graphs are tools used for many

purposes. In the simplest case one variable is plotted as a function of another, typically

using rectangular axes;

F(X) =X4-4X

Examples :

Functions of one variable :

The graph of the function.

is

{(1,a), (2,d), (3,c)}.

The graph of the cubic polynomial on the real line

is

{(x, x3 − 9x) : x is a real number}.

If this set is plotted on a Cartesian plane, the result is a curve (see figure).

Graph of the function f(x) = x3 – 9x

“ Functions of two variables “

The graph of the trigonometric function on the real line

f(x, y) = sin(x2) · cos(y2)

is

{(x, y, sin(x2) · cos(y2)) : x and y are real numbers}.

If this set is plotted on a three dimensional Cartesian coordinate system, the

result is a surface (see figure).

Graph of the function f(x, y) = sin(x2) · cos(y2).

“ Normal to a graph ”

:  , the normal to the graph is

(up to multiplication by a constant). This is seen by considering the

graph as a level set of the function  , and using

that   is normal to the level sets.

“ Generalizations ”The graph of a function is contained in a cartesian product of sets. An

X–Y plane is a cartesian product of two lines, called X and Y, while a

cylinder is a cartesian product of a line and a circle, whose height,

radius, and angle assign precise locations of the points. Fibre

bundles aren't cartesian products, but appear to be up close. There is a

corresponding notion of a graph on a fibre bundle called a section.

Vertical Line Test :

A set of points in the plane is the graph of a function if and only if no vertical line intersects the graph in more than one point.

Example :

The graph of the equation y2 = x + 5 is shown below.

By the vertical line test, this graph is not the graph of a function, because there are many vertical lines that hit it more than once.

Think of the vertical line test this way. The points on the graph of a function f have the form (x, f(x)), so once you know the first coordinate, the second is determined. Therefore, there cannot be two points on the graph of a function with the same first coordinate.

All the points on a vertical line have the same first coordinate, so if a vertical line hits a graph twice, then there are two points on the graph with the same first coordinate. If that happens, the graph is not the graph of a function.  

Characteristics of Graphs

Consider the function f(x) = 2 x + 1. We recognize the equation y = 2 x + 1 as the Slope-Intercept form of the equation of a line with slope 2 and y-intercept (0,1).

Think of a point moving on the graph of f. As the point moves toward the right it rises. This is what it means for a function to be increasing. Your text has a more precise definition, but this is the basic idea.

The function f above is increasing everywhere. In general, there are intervals where a function is increasing and intervals where it is decreasing.

The function graphed above is decreasing for x between -3 and 2. It is increasing for x less than -3 and for x greater than 2.

Using interval notation, we say that the function is

decreasing on the interval (-3, 2)

increasing on (-infinity, -3) and (2, infinity)

“DIFFERENT SHAPES OF GRAPHS ARE GIVEN BELOW “