Study graph functions

Graphs of functions f is the collection of the all ordered pairs (x, f(x)). In particular, if x is a real number, a 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 the Cartesian plane is sometimes are called 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 a surface .

Example problems for study graph functions :

Study linear functions graph:
If a function f   : R → R is defined in the form f(x) = dx + e then the function is called a linear function. Here d and e are constants.

Problem 1: 

Draw the graph of the linear function f : R → R defined by f(x) = 7x + 1.
Solution:
Draw the table of some pairs (x, f(x)) which satisfy f(x) = 7x + 1.

x	-2	-1	0	1	2
f(x)	15	8	1	-6	-13

Plot the points and draw the curve passing through these points. Note that, the curve is a straight line.



Study functions of graph:

Graph of a functions: The graph of a function f is a graph of the equation y = f(x)

Problem 2:
Draw the graph of the function f(x) =2x²

Solution:
Draw a table of some pairs (x, y) which satisfy y = 2x²

x	-3	-2	-1	0	-1	-2	-3
f(x)	18	8	2	0	2	8	18

Plot the points and draw the smooth curve passing through the plotted points.



Note:

Note that if we draw a vertical line to the above graph, it meets the curve at only one point
i.e. for every x there is a unique y

Example problem for study logarithmic graph functions :

Draw the graphs of the logarithmic functions (1) f(x) = log ₅2x (2) f(x) = log ₁₀ex (3) f(x) = log ₅3x
Sol:

The logarithmic function is the defined only for the positive real numbers. i.e. (0, ∞)
Domain: (0, ∞) Range: (− ∞, ∞)