- A function is
**even**if:- its graph is symmetric with respect to the
*y*-axis; *f*(−*x*) =*f*(*x*) always.

- its graph is symmetric with respect to the
- A function is
**odd**if:- its graph is symmetric with respect to the origin;
*f*(−*x*) = −*f*(*x*) always.

*f*(*x*) = 0.

The graph of a function can only have one *y*-intercept,
which is easy to compute:

- (0,
*f*(0)).

*f*(*x*) =*m**x*+*b*.

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,
from the Latin for ‘cutting’)
and calculate its slope:

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

For a particular function on a particular nontrivial interval, sometimes the average rate of change of that function between any two points in that interval always has the same sign (positive or negative).

- 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** (also called a *relative* extremum)
is anything that's either a local maximum or a local minimum.
(Unfortunately, different people sometimes use
slightly different definitions of these terms,
which can be clarified by adding extra adjectives.
So the local extrema here are really *strict two-sided* local extrema.
If you take another course, look at another website, or read another book,
then you may be expected to use a weaker meaning.)

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** (also called a *global* extremum)
is anything that's either an absolute maximum or an absolute minimum.
(This is the weakest and default meaning of these terms;
you're unlikely to ever be expected to use them differently.)

Note the grammar used with this terminology (both local and absolute):
when a function *f* has an extremum *at* *b*,
the extremum *is* *f*(*b*).
Calculating extrema usually requires Calculus,
so mostly you'll just look for them on the graph in this course.
(The traditional plurals of all of these ‘‑um’ words
end in ‘‑a’;
again, this comes from Latin.)

Go back to the course homepage.

This web page was written from 2011 to 2017 by Toby Bartels, last edited on 2017 April 10. Toby reserves no legal rights to it.

The permanent URI of this web page
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`http://tobybartels.name/MATH-1150/2017SP/fnprops/`

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