

The
graph of a function 
Functions
behavior, properties
and characteristic points of the graph 
Domain
and range 
Roots or zero function
values, xintercepts, yintercepts 
Increasing/decreasing
intervals
 monotony 
The
instantaneous
rate of change or the derivative 





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 
 xintercepts or zeros (roots) and
the yintercept 
 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 xaxis,
yaxis, and the origin 

Domain and range 
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 realvalued
functions of a real variable. 

Roots or zero function
values, xintercepts, yintercepts 
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
xintercept
is the point (x,
0)
where the graph of the function touches or crosses the xaxis. 
That
is, at the xintercept,
the coordinate y
= 0. 
A
zero of a function is the x
value of the xintercept.
The
zeros (roots) of a function correspond to the xintercepts
of the graph. 
The
yintercept
is the value of y
where the graph crosses the yaxis. 
The
yintercept
correspond to the point (0,
y) on
the yaxis
therefore,
at the
yintercept
the coordinate x
= 0. 

Increasing/decreasing
intervals 
A
function ƒ
is increasing on an interval if 
ƒ
(x_{1})
< ƒ
(x_{2})
for each x_{1}
< x_{2 }_{
}in
the interval. 
A
function ƒ
is decreasing on an interval if 
ƒ
(x_{1})
> ƒ
(x_{2})
for each x_{1}
< x_{2 }_{
}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 (x_{1},
ƒ(x_{1}))
and (x_{2},
ƒ(x_{2}))
of the graph of a function
is
positive, the function is increasing (going up), as shows the
figure below. 

where,
x_{2
}
x_{1}>
0 





Since
the difference x_{2
}
x_{1 }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 xaxis,
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
(x_{1},
ƒ(x_{1})), the
derivative of ƒ at
x
= x_{1 }
is the slope of the tangent line through the point (x_{1},
ƒ(x_{1})),
i.e., f
'(x_{1})
= tan
a_{t}
. 
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) 









Calculus
contents A 



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