# 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.

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