In mathematics, the graph of a function f is the collection of all ordered pairs (x, f(x)). In particular, if x is a real number, graph means the graphical representation of this collection, in the form of a curve on a Cartesian plane, together with Cartesian axes, etc. Graphing on a Cartesian plane is sometimes referred to as curve sketching. 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 its graphical representation is asurface (seethree dimensional graph).

The graph of a function on real numbers is identical to the graphic representation of the function. For general functions, the graphic representation cannot be applied and the formal definition of the graph of a function suits the need of mathematical statements, e.g., theclosed graph theoreminfunctional analysis.

The concept of the graph of a function is generalized to the graph of arelation. Note that although a function is always identified with its graph, they are not the same because it will happen that two functions with differentcodomaincould have the same graph. For example, the cubic polynomial mentioned below is asurjectionif its codomain is thereal numbersbut it is not if its codomain is thecomplex field.

To test if a graph of a curveis a function, use the vertical line test. To test if the function is one-to-one, meaning it has an inverse function, 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 . A curve is a one-to-one functionif and only if it is a function and it passes the horizontal line test.

steps and exemple :

Instructions

1

Draw a coordinate system on your graph paper using the straight edge and pencil. Draw a horizontal axis across the middle of the paper and a vertical axis down the middle.

2

Determine your domain. You can use any set of numbers as long as they don't violate the definition of a function. For example, f(x) = 1 / (x - 1) could use any number in its range except 1, which would put a 0 in the denominator of the function, making f(x) undefined.

Domain: 2 < x < 100

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3

Select a few values from the domain to plot on the graph. The more complex the function, the more points you will need to plot to capture the the curvature.

Domain f(x) = [2, 3, 4, 5, 6]

4

Input the domain values you selected into the function and calculate the associate range values. Plot the domain-range pairs on the graph using the x-axis for the domain and the y-axis for the range. For the example function f(x) = 1 / (x - 1):

Domain f(x) = [2, 3, 4, 5, 6]

Range f(x) = [1, 1/2, 1/3, 1/4, 1/5]

5

Align a French curve with the points you plotted and trace a curve that connects all the points.

  so you will draw points (2,1) and (3,1/2) and (4 , 1/3) and (5,1/4) and (6,1/5)

i hope that you understand all steps