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Let R denote the set of real
numbers, as may be represented on a
number line, and let R2 = R x R
denote a real number plane, defined
by two real number lines intersecting at the common origin (0).
Then a relation on R
is simply any collection of points in R2, and a
graph of such a relation is a visual depiction of such a set
of points,
by convention said depiction using the Cartesian coordinate system.
If the horizontal and vertical axes of the
Cartesian coordinate system
are labeled as x and y, then a relation may be represented
as {(x,y)},
a generic symbol denoting any collection of ordered pairs which
represent points in R2. In order to distinguish one relation
from
another, the usual description involves algebraic notation, either
in the form of an algebraic equation or an algebraic inequality.
Examples . . . (presented on chalkboard,
not listed here)
A function y = f(x)
is defined as a relation {(x,y)} in which it is
impossible for two different values of y to correspond to the same
value of x. That is, a function is single-valued:
plugging in one value
of x as input can give only one value of y as output.
The term "function" has been in use, with various shades of
meaning, since the 1600's.
The notation y = f(x) was first popularized by
Leonhard Euler (pronounced "oiler") in the 1700's.
The idea that a function must be single-valued did not become entrenched
until the early 1900's.
Before that time, the idea of a multi-valued function was
always allowed, that is, one in which
plugging in a single value of x might possibly generate many
different values of y. Today we
refer to such a multi-valued function as a set-valued function.
Since a function y = f(x)
is a relation {(x,y)} where unique values of y
are assigned to various values of x, it is possible to distinguish
between
the set of x-values, that is, the set of values of x for
which the function
is defined (this is called the domain of f), and
the set of y-values, which
are all the possible imaged values of y corresponding to the values
of x
in the domain of f (this set of y-values is called the range
of f).
More examples . . . (again, not listed
here)
Let G represent the graph
of a relation; that is, G is the plot of a set
of points in the xy-plane. If any vertical line (parallel to the y-axis)
crosses G in more than one point then G can
not represent the graph
of a function, since then more than one y-value corresponds to a
single x-value. However, if every possible vertical line crosses G
in
at most one point then G is the graph of some function y
= f(x).
If every possible vertical line crosses G in exactly one
point then
f is defined for all values of x, and so the domain of f
is all of R.
Note that horizontal lines are allowed to
cross the graph of a function
y = f(x) at more than one point, since there is
nothing that says the
same output y can't be obtained from different inputs x.
(What is not
allowed is different outputs from the same input.) However, if no
horizontal line crosses the graph of y = f(x) more
than once then f
is said to be one-to-one. (That is, a one-to-one function
does not
allow for the same output y to come from different input values of x).
Finally, if every horizontal line crosses the graph of y = f(x)
at least
once then the range of f is the entire set of real numbers, in which
case f is said to be onto R.
Impress Your Friends: A
fancy name for a function which is one-to-one is that it is injective,
an onto function is surjective, and a function which is both
one-to-one and onto is bijective.
Even versus Odd:
Given a function y = f(x),
if changing the sign of x automatically
also changes the sign of y, then f is called an odd
function.
Note if f is odd and if zero is in the domain of f, then
the only way
to change the sign of f(0) (since zero doesn't change sign) is for
f(0) to be equal to zero. That is, if f(0) is defined and f
is odd then
the graph of f must go through the origin (f(0) = 0).
If changing the sign of x does not
change the sign of y = f(x) then
f is called an even function. Since changing the
sign of x is the same
as reversing positive and negative on the x-axis, and since an even
function does not change if this reversal is made, the graph of any
even function must be symmetric about the y-axis. For odd functions,
where changing the sign of x also changes the sign of y
(effectively
reversing the roles of the first and third quadrants), the graph of any
odd function remains unchanged upon a 180-degree rotation about
the origin. (This is true whether or not the graph actually goes through
the origin – consider for example the graph of y = 1/x
.)
The reason for calling functions even or
odd in this manner is
because monomials with even power (such as y = x-squared)
are even functions while monomials with odd power (x-cubed)
are odd functions. Be careful, however, in rules involving addition
and multiplication of even and odd functions, which don't obey the
same rules as adding even and odd numbers. For example, the
identity function f(x) = x is odd, and the
constant function g(x) = 1
is even. The product function f · g is odd (since x ·
1 = x), contrary
to the rule from arithmetic that odd times even equals even, and the
sum (f + g)(x) = x + 1 is neither odd nor
even.
Again, even more examples not illustrated
here —
(Maybe if the Math Dept can pony up to have some useful software installed
on this
here lecture presentation machine then next year's demonstration will be a
bit better.)
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