Algebraic Functions: Concepts & Procedures
Examples
of Algebraic Expressions. Algebraic
expressions are constructed from polynomials by adding, subtracting,
multiplying, dividing, raising to an integer power, and taking
roots. Examples include: x^{1/2},
3 x^{5/3} – 17,
–x^{–1} + 7, and
x^{2}– x^{–2}– 4.
Algebraic functions. Algebraic
functions are expressions that involve only the algebraic
operations of addition, subtraction, multiplication, division,
raising to an integer power, and extracting an odd number root
or even number root. Strictly speaking, any polynomial or rational
function is also an algebraic function.
The absolute value function, abs(x)=x,
is of special interest. Since it can also be expressed as (x^{2})^{1/2},
it is an "algebraicallygenerated" function (although, for technical reasons, it is classified as a transcendental function).
n^{th} Roots
of a number. An n^{th} root
(n is a positive integer) of a given number is a number
that, when raised to the n^{th} power
(that is, when n copies of an n^{th} root
are multiplied), produces the given number. There are n distinct n^{th} roots
(DeMoivre's Theorem) of any nonzero number, and they may be real
or imaginary numbers. No matter how large n is, at most
two of the n^{th} roots
are real, and when there are two real roots, they are of opposite
sign.
Odd
roots of a number. If n ≥ 3 is
odd, there is exactly one real root of a given (real) number.
For example, the three cube roots of –1 are: –1
and (1/2)(1±3^{1/2}i),
where i is
the imaginary unit, i=(–1)^{1/2}.
When graphing functions with odd roots in the xy plane,
plotting is restricted to the realvalued ycoordinates.
Below are the graphs of the cube root, fifth root, and seventh
root functions. All go through the points (–1,1), (1,1) and
the origin, (0,0). They are defined for all values
of x.
Even
roots of a number. If n ≥ 2 is
even and a given (real) number is positive, there are two real
n^{th} roots of the
number, numerically the same but opposite in sign. For example,
the four 4^{th} roots
of 1 are ±1 and ±i . If n ≥ 2 is
even and a given (real) number is negative, all n^{th} roots
are imaginary. For example, the four 4^{th} roots
of –1 are (2^{1/2}/2)(±1±i). Below
are the graphs of the square root, fourth root, and sixth root
functions. All go through the points (1,1) and the origin, (0,0).
They are only realvalued for nonnegative values of x.
For that reason, their graphs only appear in the right
halfplane x≥0.
Complete Graph. A
complete graph is one that includes all the important mathematical
features. For algebraic functions, this means the viewing window
should be just large enough to display all x and yintercepts,
all maxima and minima, all points of inflections, all holes, all
2hole jumps, all vertical asymptotes, all horizontal asymptotes,
all vertical tangents, all vertical cusps, all corners, and all isolated
points. When you enter an algebraic function and click the GraphIt button, WebGraphing.com automatically
displays a complete graph. Armed with this knowledge, you can select
different viewing windows to zoom in on particular
aspects of the graph or refine it more to your liking.
Zoom
In/Out (Resizing the Viewing Window). If you wish to
graph again with a different display window, you can do this
in two ways: (1) you can enter the lower and upper xbounds
only and let WebGraphing.com determine optimal ybounds
(Smart Zooming) that include all the important
mathematical features contained within your xbounds,
or (2) you can enter both the lower and upper x and ybounds
to choose the precise viewing window. The first option is especially
useful if you are zooming in on a peak or valley or a change
in concavity, since in that case WebGraphing.com chooses
an optimal range of yvalues. The choice of an appropriatelychosen
display can be important to visually verify, beyond the indicated
colorcoding, underlying features like maxima, minima, and points
of inflection.
Domain and Intercepts
Domain. Algebraic
functions involving even roots may require restrictions to their
domains in order to produce real yvalues. For example,
(x–1)^{1/2} is real
valued only when x ≥ 1.
Also, any xvalue producing zero in the denominator of
an algebraic function must be excluded, as in ((1–x)^{1/2})/x, where x≠0
since division by 0 is not defined. The domain of an algebraic
function is the largest set of real xvalues that produce
real yvalues.
Horizontal
and Vertical Asymptotes
(What
are Asymptotes?)
Horizontal
Asymptotes. A horizontal
asymptote is a horizontal line, y=c,
which the graph of a function approaches either as x tends
to +∞ and/or as x tends to
−∞. This
is customarily indicated on the graph by plotting the horizontal
asymptote as a dashed horizontal line. Unlike vertical asymptotes,
the curve representing the algebraic function may touch or cross
a horizontal asymptote. While rational
functions may have at most one horizontal asymptote, algebraic
functions may have up to two horizontal asymptotes.
Vertical
Asymptotes. A vertical asymptote is a vertical
line, x=c, which the graph of an algebraic
function approaches but never touches. For algebraic functions,
vertical asymptotes always occur at the zeros of the denominator
that are not eliminated by improper cancellation. As x approaches
such a denominator zero through domain values from either one
of the left or right sides, the yvalues of the function
tend to either +∞ or −∞. This
is customarily indicated on the graph by plotting the vertical
asymptote as a dashed vertical line. The curve never intersects
the dashed vertical line, since the defining algebraic function
does not have any yvalue at the denominator zero. The value x=c of
a vertical asymptote is sometimes referred to as an essential
singularity, since no redefinition of the function at that xvalue
can ever make it continuous (for contrast, see holes below).
Holes
and Jump Discontinuities
Holes. When an
algebraic function has a common factor in the numerator and denominator
that can be eliminated in the denominator by improper cancellation,
the importance of this lies in the fact that the algebraic function
is not defined at the zero of the common factor since it appears
in the denominator. Any common factor that is eliminated in the
denominator due to improper cancellation does not give rise to
a vertical asymptote. This is traditionally signified on the graph
by having an empty circle appear at the one or two places where
the function might otherwise be continuous from at least one side.
When the limit from the left is finite and equals the limit from
the right, this is called a hole and is sometimes referred
to as a removable singularity (or removable discontinuity),
since, unlike a vertical asymptote, the algebraic function could
be redefined to be continuous at that point.
Jump Discontinuities. When
an algebraic function is undefined at a point, say, for example x/x
at x=0, but the limits from the left side and
right side are finite but unequal (in this example, as x tends
to 0 from the left, x/x tends
to –1; as x tends to 0 from the right, x/x tends
to +1), this gives rise to a jump discontinuity. This
is traditionally signified on the graph by having an empty circle
appear at the two places where the function has limits. Here is
an example of an algebraic function with a jump discontinuity.
Vertical Tangent Lines, Vertical Cusps, and
Corners
Vertical Tangents and Vertical Cusps. If
a function f is differentiable at x=a, then
the graph of f has a tangent line at a. What
happens when f is not differentiable? The graph of y=f (x) is
said to have a vertical tangent line at (a, f (a)) if:
(1) f is continuous at a,
(2) f '(a) does not exist (as a real
number), and (3) f '(x) approaches one
of either ∞ or −∞ as x approaches a. An
example of this is y=x^{1/3} at (0,0). Note
that if x=a is an endpoint of an interval in a
restricted domain, f is said to have a vertical tangent
as determined from one side only; in that case, the continuity
of f and the limit of f '(x) are
interpreted as x approaches a from one side only.
An example of this is y=x^{1/2} at when a=0.
As x approaches 0 from the right, f '(x) approaches ∞.
Geometrically, the secant lines through (0,0) and (x,x^{1/2})
approach vertical as x approaches 0. In the case where a is
an interior point of the domain, if the approach of f '(x) is +∞ as x approaches a from
one side and −∞ as x approaches a from
the other side, a sharp point occurs that is called a vertical
cusp. An example of this is y=x^{1/3} at (0,0).
Corners. The graph of y=f (x) is
said to have a corner at (a, f (a)) if:
(1) f is continuous at a,
(2) f '(a) does not exist, and (3) f '(x) approaches
a real number as x approaches a from the left
side and f '(x) approaches a different
real number as x approaches a from
the right side. An example of this is the absolute value function, abs(x)=x, at (0,0), where f '(x) approaches –1 as x approaches
0 from the left side and f '(x) approaches
+1 as x approaches 0 from the right side.
Maxima and Minima
Local Maxima. The yvalue f (c)
is a local maximum value (also called relative maximum
value) of f if there is an open interval containing
the xvalue c (halfopen if c is an
endpoint in the domain of f ) where f (c) ≥ f (x). For
interior points in the domain of f, the First Derivative
Test states that when the graph of the function f,
continuous at x=c, is increasing on
the immediate left of the number x=c and decreasing on
the immediate right of the number x=c, then
the value of f at c is locally the
largest, i.e., f (c) is
a local maximum. This test can be extended to endpoints
in the domain of f: if x=c is
a left endpoint and decreasing on the
immediate right, or if x=c is a right
endpoint and increasing on the immediate
left, then f (c) is a local
maximum.
Local Minima. The yvalue f (c)
is a local minimum value (also called relative minimum
value) of f if there is an open interval containing
the xvalue c (halfopen if c is an
endpoint in the domain of f ) where f (c) ≤ f (x). For
interior points in the domain of f, the First Derivative
Test states that when the graph of the function f,
continuous at x=c, is decreasing on
the immediate left of the number x=c and increasing on
the immediate right of the number x=c, then
the value of f at c is locally the
smallest, i.e., f (c) is
a local minimum. This test can be extended to endpoints
in the domain of f: if x=c is
a left endpoint and increasing on the
immediate right, or if x=c is a right
endpoint and decreasing on the immediate
left, then f (c) is a local
minimum.
Determining
Local Maxima and Local Minima.Any value
of x in the domain of f is called a critical
number of f ' (also called critical point or critical
value) if either f '(x)=0 or f '(x)
does not exist. For continuous functions, the local
maxima and local minima can only occur at the critical numbers
or endpoints of the domain of f. These numbers separate
the domain of f into intervals. To determine all local
maxima and local minima, select a convenient "Test Value" on
each such interval and determine the sign of f '(Test
Value). At each critical number or endpoint,
there are three possibilities.
1. If the sign of f '(Test Value) is
positive (+) on the left side (interval to the left)
of a critical number of f ' and negative (−) on
the right side, suggesting visually that the function is increasing on
the left side and decreasing on
the right side, then f has
a local maximum at that critical number. If the sign of f '(Test
Value) is positive (+) on the left side
of a right endpoint (suggesting visually that the function is increasing on
the left side) or negative (−) on the right
side of a left endpoint (suggesting visually that the function
is decreasing on the left side),
then f has a local maximum
at that endpoint.
2. If the sign of f '(Test Value) is
negative (−) on the left side of a critical number
of f ' and positive (+) on the right side,
suggesting visually that the function is decreasing on the left
side and increasing on the right side, then f has
a local minimum at that critical number. If the sign of f '(Test
Value) is negative (−) on the left
side of a right endpoint (suggesting visually that the function
is decreasing on the left) or negative (−) on the
right side of a left endpoint (suggesting visually that the function
is increasing on the right side), then f has
a local minimum at that endpoint.
3. If there is no change in sign of f '(Test
Value) from either side of a critical number to
the other side, then the critical number is not a local maximum
or local minimum for f.
In the latter case, if the sign of f '(Test
Value) is (+) on the left side of a
critical number and (+) on the right side, this
suggests visually that the function is increasing on the left
side and increasing on the right side. So, the curve has a
horizontal tangent at that zero but the point of tangency is
not a turning point. Alternatively, if the sign of f '(Test
Value) is (−) on the left side
of a critical number and (−) on the right
side, this suggests visually that the function is decreasing
on the left side and decreasing on the right side. Once again,
the curve has a horizontal tangent at the critical number but
the point of tangency is not a turning point. In both of these
cases, the horizontal tangent cuts through to the other side
of the curve. Visually, this suggests a change in concavity,
which is treated using the second derivative.
Increasing/Decreasing Intervals
Increasing. Geometrically
speaking, a function f is increasing on an interval
if the function is rising over the interval as you look at the
graph from left to right. By definition, f is increasing on
an interval I if f (a) < f (b) whenever
a < b, for a, b in I. This
is true if f '(x) > 0 at
interior points of the interval I. The latter condition
is adequate (sufficient) to establish that a function is increasing,
but it does not always work to verify the “increasing” property,
as indicated by the example of the algebraic function
f (x)=x^{1/3}, where
f ' does
not exist at x=0 but y=f (x) is
increasing on any interval containing 0. The increasing segments (also
called arcs) of a graph are shown on WebGraphing.com in red.
Decreasing. Geometrically
speaking, a function y=f (x) is decreasing on
an interval if the function is falling over the interval as you
look at the graph from left to right. By definition, y=f (x)
is decreasing on
an interval I if f (a) > f (b)
whenever a < b, for a, b in I.
This is true if f '(x) < 0 at
interior points of the interval I. The latter condition
is adequate (sufficient) to establish that a function is decreasing,
but it does not always work to verify the “decreasing” property,
as indicated by the example of the algebraic function f (x)=–x^{1/3},
where f ' does
not exist at x=0 but y=f (x) is
decreasing on any interval containing 0. The decreasing segments
of a graph are shown on WebGraphing.com in blue.
Determining Increasing and Decreasing Intervals. To
determine the intervals of increase and decrease for an algebraic
function f, first find all the critical numbers of f. These
numbers separate the domain of f into intervals. Select
a convenient Test Value on each such interval and determine
the sign of f '(Test Value). If the sign
is positive (+), then the function is increasing on the interval. If
the sign is negative (−), then the function is decreasing on
the interval. Based on these considerations, the intervals
of increase and decrease can be determined by creating a table with
critical numbers of f and test values.
Points of Inflection. A
point on a graph where the function is continuous and there is
a change in concavity is called a point of inflection. Any
point where the function is not continuous cannot be a point of
inflection.
Determining
Points of Inflection. Any value of x in
the domain of f is called a critical number of f ''
if either f ''(x)=0 or f ''(x)
does not exist. For any continuous function,
a point of inflection can only occur at a realvalued zero of f ''. For
continuous functions, the points of inflection can only occur at
the critical numbers in the domain of f. These numbers
separate the domain of f into intervals. To determine
all points of inflection, select a convenient "Test Value" on
each such interval and determine the sign of f ''(Test
Value). At each critical number of f '', there
are three possibilities:
1. If the sign of f ''(Test Value) is
positive (+) on the left side (interval to the left) of a critical
number of f '' and negative (−) on
the right side (interval to the right), indicating visually that
the function is concave
up on the left side and concave
down on the right side, then f has
a point of inflection at that critical number.
2. If the sign of f ''(Test Value) is
negative (−) on the left side of a critical number of f '' and
positive (+) on the right side, indicating visually
that the function is concave down on the left side and concave
up on the right side, then f has
a point of inflection at that critical number.
3. If there is no change in sign of f ''(Test Value)
from one side of a critical number of f '' to
the other side, then the zero is not a point of inflection for f.
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United States Patent Numbers 7,432,926, 7,595,801, & 7,889,199.
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