Functions of a Graph

A rule that assigns one element y in Y to each element x in X is called a function f from a set of elements X to a set of elements Y .A function can also be represented as a set of ordered pairs (x, y) in which there is only one y-value for each x-value in the set. This means that no x-values with distinct y-values can be repeated.

For example : F = { (1,2),(3,4),(5,6)} is a function while G = {(1,2),(2,3),(2,4),(4,5)} is not a function.

The domain of f for a function f: X Y is the set X.

When we express a function as a collection of ordered pairs, this corresponds to the set of x-values (x, y).

The range of a function f: X Y is the set of y-values that make y = f(x) for some x in X.

When we express a function as a collection of ordered pairs, this corresponds to the set of y-values (x, y). The range of the function y = x is all real y 0

Types of Graphs :

• Identity Function

• Constant Function

• Polynomial Function

• Modulus Function

• Signum Function

• Rational Function

• Greatest Integer Function

Identity Function :

For each x R, consider the real-valued function f: R R by y = f(x) = x (i.e. the set of real numbers). The identity function is a type of function like this. This function f’s domain and range are also R. As seen in the diagram below, the graph of this type of function is a straight line flowing through the origin.

Constant Function :

The function f: R R by y = f (x) = c, x R is called a constant function since c is a constant and each x R is a constant function. This function’s domain is R, and its range is c. The graph of a constant function, y = f(x) = 2, however, is shown below.

Polynomial Function : 

For each x in R, y = f(x) = a0 + a1x + a2x2 +…+ anxn, where n is a non-negative integer and a0, a1, a2,…,an R is said to be a polynomial function. A parabola is the graph of this type of function.

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Intercepts and Points : 

The place where the coordinate axes cross is known as the intercept. Put y = 0 in the equation of the curve to identify the spots where it cuts the x-axis. Similarly, in the equation of the curve, put x = 0 to identify the locations where the curve cuts the y-axis. If we put y = 0 in the function y = (x + 5)2(x+6) we get x = -5 & -6 and if we put x = 0 we get y = 60. As a result, the intercepts are (-5, 0), (-6, 0), and (0, 60).. This is also known as the function’s zeros, and factorization is done using quadratic and other more difficult formulas. To locate the points on the polar curve, we create a table of r and locate their corresponding points.

Slope of Curve at a Point :

Tangent is the line that intersects the curve exactly at one point. The slope of the curve at any point P is the same as the slope of the tangent line at that point. The rate of growth or decrease of the curve as it passes through P is measured by this tangent line at P. If there is no constant term in the equation, the curve passes through the origin.

Symmetry :

Even symmetry means that a function’s graph is symmetric when viewed from the y-axis. • Odd symmetry: A function’s graph is symmetric when rotated 180 degrees around the origin.

Conclusion :

The independent variable (commonly written as x) is the first variable, while the dependent variable (typically written as y) is the second. It’s a rule that explains how one quantity affects the other. So, a function is a set of rules that are applied to an input to produce a result. The number or value entered into a function is called the input. The function’s output is the number or value it returns. The collection of all points whose coordinates (x, y) fulfil the function y = f is known as the graph of a function (x). This signifies that there is a matching y-value for each x-value produced by substituting into the supplied function.