Properties of functions
Last week, we looked at properties of graphs in the plane,
which we can now think of as properties of relations.
Since we can think of functions as a special kind of relations,
these properties also apply to functions.
However, since functions had a very different historical development,
all of the names are different now!
Symmetry: even and odd functions
As graphs can be symmetric with respect to an axis or the origin,
so a function can be even or odd.
A function's graph is hardly ever symmetric with respect to the x-axis.
- A function is even if:
- its graph is symmetric with respect to the y-axis;
- f(−x) = f(x) always.
- A function is odd if:
- its graph is symmetric with respect to the origin;
- f(−x) =
Intercepts: roots and zeroes
The graph of a function can only have one y-intercept,
which is easy to compute:
There is no special name for f(0).
There are sometimes shortcuts for this, but they're hardly worth the bother.
To find the x-intercepts,
you must solve the equation
The solutions to this equation are the roots of f.
The book calls these the zeroes of f,
but I find that term confusing.
There are many shortcuts for this for certain kinds of functions,
and they're often very helpful.
Slopes: rates of change
A linear function has the form
Its graph is a line with slope m and y-intercept (0,b).
When talking about the function,
we call this number m the rate of change of f.
The slope of a vertical line is not defined, but that's OK,
since a vertical line is not the graph of a function.
Most functions are not linear.
However, given two inputs a and b of the function
(and assuming that the function is defined between a and b),
we can imagine a line drawn through the corresponding points on the graph
(which is called a secant line)
and calculate its slope:
This number (the slope of that secant line)
is the average rate of change of f
from a to b.
- m =
[f(b) − f(a)] ÷
[b − a].
For a particular function on a particular interval,
sometimes the average rate of change of that function
between any two points in that interval
always has the same sign.
Mostly you will just look for these on the graph.
- The function is increasing on the interval
if the average rate of change is always positive.
- The function is decreasing on the interval
if the average rate of change is always negative.
- The function is constant on the interval
if the average rate of change is always zero.
A function has a local maximum at an input b
if it is increasing on an interval [a,b]
and decreasing on an interval [b,c].
It has a local minimum at b
if it is decreasing on an interval [a,b]
and increasing on an interval [b,c].
A local extremum
is anything that's either a local maximum or a local minimum.
When a function f has a local extremum at b,
the local extremum is f(b).
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This web page was written in 2011 by Toby Bartels,
last edited on 2011 January 18.
Toby reserves no legal rights to it.
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