Functions behavior, properties and characteristic points of the graph
Domain and range
Roots or zero function values, x-intercepts, y-intercepts
Increasing/decreasing intervals, monotony
The instantaneous rate of change or the derivative
Continuity and discontinuity
The graph of a function
The graph of a function ƒ is drawing on the Cartesian plane, plotted with respect to coordinate axes, showing functional relationship between given variables containing all those points (x, ƒ (x)) which satisfy the given relation.
The points lying on the curve satisfy the relation that forms the shape of the graph.
The graphic representation of a function provides insight into the behavior of the function.
Functions behavior, properties and characteristic points of the graph
To sketch the graph of a function we should know its properties and find out its characteristic points, as
- domain and range
- x-intercepts or zeros (roots) and the y-intercept
- intervals of increasing and decreasing
- continuity and discontinuity
- vertical, horizontal and oblique or slant asymptotes
- turning points (extremes, local or relative maximums or minimums)
- inflection points and intervals of concavity
- symmetry (odd and even functions) with respect to the x-axis, y-axis, and the origin
Domain and range of a function
The domain is the set of values of the independent variable of a given function, i.e., the set of all first members of the ordered pairs (x, ƒ (x)) that constitute the function.
The range is the set of values that given function takes as its argument varies through its domain. It is the image of the domain.
The codomain is the set within which the values of a function lie, as opposed to the range, which is the set of values that the function actually takes.
Therefore, the range must be a subset of, but may or may not be identical with the codomain.
We will only consider real-valued functions of a real variable.
Roots or zero function values, x-intercepts and y-intercepts
A zero of a function is a value of the argument of a function at which the value of the function is zero.
An intercept is the point at which a given function intersects with specified coordinate axis, or the value of that coordinate at that point.
An x-intercept is the point (x, 0) where the graph of the function touches or crosses the x-axis.
That is, at the x-intercept, the coordinate y = 0.
A zero of a function is the x value of the x-intercept. The zeros (roots) of a function correspond to the x-intercepts of the graph.
The y-intercept is the value of y where the graph crosses the y-axis.
The y-intercept correspond to the point (0, y) on the y-axis therefore, at the y-intercept the coordinate x = 0.
Increasing or decreasing intervals of a function
A function ƒ is increasing on an interval if
ƒ(x1) < ƒ(x2)  for each x1 < x in the interval.
A function ƒ is decreasing on an interval if
ƒ(x1) > ƒ(x2)  for each x1 < x in the interval.
By looking at the graph of a function being traced out as the value of the input variable x increases from left to right then, if at the same time the output value y = ƒ (x) also increases, we say the function is increasing.
If the output value decreases as x increases, then we say the function ƒ is decreasing.
Thus, if the slope or gradient m of the secant line passing through the points (x1, ƒ(x1)) and (x2, ƒ(x2)) of the graph of a function is positive, the function is increasing (going up), as shows the figure below.
 where,   x2 - x1> 0

Since the difference x2 - x1 is always positive, when the function is decreasing (going down), the slope will be negative.
The instantaneous rate of change or the derivative
The ratio of the rise and the run, called the difference quotient, that equals the value of the tangent of the angle between the direction of the secant line and x-axis, becomes the slope (gradient) of the tangent line as the difference Dx tends to zero, and is called the instantaneous rate of change or the derivative at the point of the function.
For a given function ƒ and point (x1, ƒ (x1)), the derivative of ƒ at x = x1 is the slope of the tangent line through the point (x1, ƒ(x1)), i.e.,  f '(x1) = tan at .
The gradient of a curve at a point on its graph, expressed as the slope of the tangent line at that point, represents the rate of change of the value of the function and is called derivative of the function at the point, written
y' = dy/dx =  f '(x).
Continuity and discontinuity of function
A function that has no sudden changes in value as the variable increases or decreases smoothly is called continuous function.
Or more formally, a real function  y = ƒ (x) is continuous at a point a if the limit of ƒ (x) as x approaches a is ƒ (a).
If a function does not satisfy this condition at a point it is said to be discontinuous, or to have a discontinuity at that point.
Functions contents C