# Properties of functions (§3.3)

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 a function as a special kind of relation,
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 a graph can be symmetric with respect to an axis or the origin,
so a function can be even or odd.
- 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*) =
−*f*(*x*) always.

A function's graph is hardly ever symmetric with respect to the *x*-axis.
## Intercepts: roots and zeroes

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.
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.
## 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.
Few functions are 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:

*m* =
[*f*(*b*) − *f*(*a*)] ÷
[*b* − *a*].

This number (the slope of that secant line)
is the **average rate of change** of *f*
from *a* to *b*.
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.

- 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.

Mostly you will just look for these on the graph,
rather than calculating rates of change.
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.
Warning:
When a function *f* has a local extremum *at* *b*,
the local extremum *is* *f*(*b*).

A function has an **absolute maximum** at an input *b*
if *f*(*a*) ≤ *f*(*b*)
for any *a* in the domain of *f*.
It has an **absolute minimum** at *b*
if *f*(*a*) ≥ *f*(*b*)
for any *a* in the domain of *f*.
An **absolute extremum**
is anything that's either an absolute maximum or an absolute minimum.
Calculating these usually requires calculus,
so mostly you'll just look for these on the graph too.

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This web page was written from 2011 to 2013 by Toby Bartels,
last edited on 2013 October 17.
Toby reserves no legal rights to it.
The permanent URI of this web page
is
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