|
YOU are better than YOU think. Show
yourself how:
|
// _ _ \\
/\ /\
<| (o) (o) |>
\ | | /
|
Read logic
chapters 1 to 5 in online volume Three
Skills for Algebra for greater skills & confidence
in work
and study.
Read notes and textbooks like a lawyer, so that no nuance, no subtlety and no clause escapes your attention. |
-/[]\-
||
/ \_
||||||||||||||||||||||||||||
Logic
chapters 1 to 5 re- appear not in sequence, as is or longer,
in Volume 1A, Pattern Based
Reason, Bon Appetite.
Logic
Mastery
Amazing, Amusing, Amorous, Delicious, Delightful, Edifying,
Strengthening Elixir.
It eases work & learning difficulties Makes the hard easier. Opens eyes.
Leads to greater precision.
in reading and
writing
Logic
mastery makes the hard, easier. Logic
mastery leads to better, stronger and richer comprehension. Logic
mastery improves reading and writing. Logic
mastery ease learning difficulties. Logic
mastery gives a headstart. In sum, logic
mastery will develops critical thinking, improve reading and writing,
and give a firmer base for work and studies at many levels. Good luck.
After logic,
(a) continue reading Three
Skills for Algebra, chapters 8 to 14 and do so alongside site area on solving
liinear Equations ; or (b) see this calculus
starter lesson and Volume 3, Why
Slopes & More Math, chapters 2 to 6;
|
// _ _ \\
/\ /\
<| (o) (o) |>
| |
| |
\
/
\ = /
|
Caution: Site advice is approximately
correct, for some circumstances, not all. That leaves room for thought |
-/[]\-
||
_ / \
||||||||||||||||||||||||||||
What may be learnt and when depends on how skills
and concepts are developed. Making the hard easier and clearer will allow
earlier & richer development of skills and concepts.
Try the Twiddla
Whiteboard. In principle, it allows
to people to draw and chat together online on a copy of this webpage or a clean
sheet. The chat may be via text or audio. Visit www.twiddla.com
to set up whiteboards to work with the webpage of your choice.
For online automated help in senior high school maths & calculus,
visit quickmath.com For Automatic
Calculus and Algebra Help with derivatives, integrals, graphs, linear equations,
matrix algebra, visit calc101.com
With overlap, each site quickmath
& calc101offers a different range of
services, some free, some not, all based on webmathematica. Good luck.
| |
Functions and Relations
In modern pure mathematics, functions and the allied concept of
relations are identified with sets of ordered pairs. That provides the
eventually useful, set-theoretic viewpoint. Yet before it you should meet and
understand the previous, broader and impure dependency viewpoint.
Logical or Pedagogical Preparation (Pre-requisites)
-
Word have been missing or use unclearly in
mathematics. The introduction of the notion of what is a variable and
a quick review of three skills for algebra, the use of notation in
mathematics, and the forward and backward use of formulas in chapters 8 to
14 in Volume 2, Three Skills for
Algebra, might fill gaps in the comprehension of algebra, and
develop the algebraic maturity needed pedagogically if not logically for the
current study of functions and relations.
-
While Professors of Mathematics Education
advocate the greater use of calculators, courses in calculus and
senior secondary school courses still require students to master and
understand exact arithmetic with fractions without a calculator and the use
of prime numbers and factorization, material that appears in earlier
courses. The site area .Solving
Linear Equations introduction of stick diagrams can be reviewed
(despite opposition) so that students may visualize and consolidate some
fraction skills and concepts. The site area in full provides students and
teachers a model, a lower bound, for the solution of linear equations from
one equation in one unknown to systems of n equation in n unknowns where n =
2, 3 or 4. The site area Solving
Linear Equations in covering a simpler topic also develops a greater
algebraic maturirty,needed pedagogically if not logically for the current
study of functions and relations.
-
For students who have met slopes and/or
polynomials before the discussion of functions, the Geometric
and Algebraic previews of
calculus will provide motivation for the study of slopes (why slopes) and
for the factorization of polynomials. The algebraic previews will
develop more algebraic skills and concepts, and still greater algebraic
maturity needed pedagogically if not logically for the current study of
functions and relations. These examples may be woven into the
monoticity analysis discussion of on what intervals, real-valued
function y = f(x) of a single real variable x are increasing or decreasing.
and what intervals those functions are positive, negative or zero. A
point is given by a very short interval.
Column I
Functions Before Sets
(Cover First)
|
Column II
Functions with a
Set-Theoretic Focus
|
Methods to Define Computation and Assignment Rules :
- Using
Formulas (with use of function notation to indicate dependence of
one number or quantity on several others. (math 436)
- Using Arrow
Diagrams, Tables and Sets of Ordered Pairs (listed or plotted) -
functions with finite domains. (math 436)
- Using
Curves and Infinite Sets of Points in the Plane - When the
vertical rule holds, a set of points or curve in the plane can be used
to define a function f(x) via the vertical line method.
Note: Graphing a function f gives a set of points or curve in
the plane for which the vertical line method for computing a function
yields the same function f. (math 436)
- Functions with Infinite
Domains - a few exercises (math 436)
- Properties of Functions: or Definitions & Examples to
introduce and describe: Domains, Ranges, Injectivity (1 to
1) or not (many to 1), Onto or Surjectivity, Monoticity - where are
real value functions of a single real variable increasing, decreasing
or constant. Tools: Interval notation and symbols for there exists
and for all. More examples given by calculus preview -
geometric & algebraic (Material for calculus, if not an enriched
436 or 536)
- Sign, Zero
and Monoticity Analysis - Four Geometric Starter Lesson or
Exercises builds algebraic thinking skills.
|
Curve or Set Viewpoint of Functions and Relations
In the foregoing examples, you have seen sets appear in the description
of the domains and ranges of functions, and in the definition of function
using sets of ordered pairs. The latter implies or suggest the Set
Based View and Codification of what is a function in site pages with
the following ideas. (Here are more ideas for math 436).
- Set Existence
and Construction (technical starting point)
- Interval Notation.
Next (?) see Domains and ranges for a
zoo of functions using interval notation.
- Assignment and Computation
Rules without & then with ordered pairs.
- Concept of
a Relation, a Set-Based Codification and Generalization.
- Why call a set of
ordered pairs a relation? Numerical Exercise Included.
- Source, Target, Domain and
Range Set for functions and relations - plus Definition of
subjection, injections and bijections - set viewpoint
- Injectivity of Real Valued
Functions - injectivity, one-to-one, two-to-one, many-to- one, or
not one- to-one.
- Sign Analysis, Zero
Analysis, Where are functions positive, negative or zero?
- Monotonicity Analysis: Where
are functions increasing, decreasing etc.Why strictlyincreasing and
strictly decreasing functions are one to one, that is, injective.
.
- Extrema or Max-Min Analysis
Where do they have their greatest and least values. What are minima
and maxima.
- Exercises with
Formulas and Graphs - Numerical Experience (!)
- Domains and ranges for a zoo of
functions using interval notation.
- The absolute Value
Function (Qc math 536)
- Functions Revisited (for
teachers, if not students)
|
Backward Use of Functions
Or, Inverse Functions and Their Definition
The set or curve in the plane viewpoint (Route 2) has
advantages in discussing the backward use of formulas y = f(x) where instead
of calculating or obtaining y from x as in the forward use, we try to
obtain x from y. Remember that when you meet the discussion of inverse
functions.
A curve in the plane may be regarded as a set of points or ordered
pairs. The graph of a function f even when the function f or f(x) is
introduced by other means may be used for calculation of y = f(x), that is
the forward use of the function, and for the backward question of how x
depends on y when y = f(x), y is given and x is to be computed. This
backward question provides a context for the following.
- Using
the Horizontal Line Method - Part I . Just as there is a vertical
line method for defining and calculating functions, there is also a
horizontal line method. Here if S is a set of points for which the
horizontal line method can be used to compute a function y = f(x) then there
is a twist, the graph of the function f
graph (f) = { (a,b) | (b,a) belongs to S}
is equal to a "transpose" of the set S in which the first and
second coordinates are swapped. (Secondary V Subject). Note: In
earlier courses, reflection of a point (a,b) across the line y =
x gives the point (b,a) with coordinates interchanged or swapped.
- Using the
Horizontal Line Method Part II. If we apply the horizontal method
to all or part of the graph of a function y = f(x) we may obtain another
function h such that z = h(y) implies y = f(z), and perhaps, vice-versa.
(Needed for discussion of inverse trig functions in calculus or secondary V
mathematics)
- Several more ways to
define Functions - A brief glimpse of the future if you are in secondary
IV or V, and glimpse of the present or past if you are studying
calculus.
- Algebraic Calculation of Inverse Functions. Suppose y = f(x)
where f(x) is a function given by a formula of some type. The inverse
function
f--1(x) = g(x)
if it exist, should have the property that g(f(x)) = x for each x in domain
of f and also f(g(w)) = w for each w in the range of the original
function f. Now f(y) = x may imply y = h(x) for some unique function
h(x) or it may give more than one formula or solution h(x) for y. In
the latter case, the function f is not one to one. In the former case,
f is one to one, f--1 exists,
and f--1(x) = h(x).
Proof that f--1(x) = h(x).
: If x = f(h(x)) then by substitution f--1(x)
= f--1( f( h(x)) = f--1(f(y))
= y = h(x)
Remark: If f(y) = x implies an equation linear in y (with the
y coefficient nonzero) then y will be uniquely determined. If f(y) = x
implies an equation quadratic in y (or more generally with a polynomial
dependence on y) then their could 2 or more formulas h(x) for y, one formula
per real root of a quadratic or more general polynomial in y.
The foregoing lessons provide a basis for defining inverse trigonometry using
parts of the graphs of trig functions - the restriction of the latter to
intervals to obtain functions that are one-to-one (invective). The twist,
reflection across the line y = x in the Cartesian plane, connects the graph of a
function and the graph of its inverse. And in calculus, the area
under the curve definition of the natural logarithm leads to a one-to-one
function. Its inverse is the exponential function.
| |
www.whyslopes.com
Analytic Geometry
& Functions, etc
Area Entrance
FNs & Dependency
FN With Finite Sets
FN Vertical Line Rule
FN Infinite Domains
FN Sets-Theory
FN Interval Notation
FN: Sets - Continued
(FN) Sets & Relations I
(FN) Relations & Sets
FN Source Target Domain Range
(FN) Injective or Not
(FN) Sign & Zero Analysis
(FN) Increasing/Decreasing
(FN) Extrema
FN Numerical Exercises
FN Step Sawtooth Abs.Value
FN Horizontal Line Rule
FN Inverse Functions
FN Many Ways to Define
Entrance + Pages Below this page
Pages at Current Level (C) Complex Numbers
(FN) What are Functions?
(FN) Functions - More
SZM: Sign Analysis
(L) Lines Summary
(P) Polynomials (*,+,-)
(Q) Quadratics
(D) Simplify Square Roots
(T) Unit Circle Trig
Conic Sections
Links
More Links
Area Entrance
Equal Sign Use/Abuse
Real Numbers
Say More Positive
Linear Inequalities
Triangle Inequality
Absolute Value |x|
|x| Eq'ns & Inequalities
Rectangular Coords
Shortest Path
Distance Formulas
Add & Multiply Points
Polar Coordinates
Radians
(A) Vectors
(A) Coordinate Arithmetic
(A) Navigation on Maps
(A) Addition Geometrically
(A) Rotation
(PT) Translations
(PT) Dilatations
PT: Rotations
Pages at Above this Page
Extras: Not all perfect.
Equal Sign Use/Abuse
Real Numbers
Say More Positive
Linear Inequalities
Triangle Inequality
Absolute Value |x|
|x| Eq'ns & Inequalities
Rectangular Coords
Shortest Path
Distance Formulas
Add & Multiply Points
Polar Coordinates
Radians
(A) Vectors
(A) Coordinate Arithmetic
(A) Navigation on Maps
(A) Addition Geometrically
(A) Rotation
(PT) Translations
(PT) Dilatations
PT: Rotations
Links to Site Pages outside this site area follow - co- and pre-
requisites.
Road
Safety Message
Easy Consequences of this (newest) Complex
Number. Starter Lesson follow below to provide an alternate
development of HS or college maths.
Vec & Cmplx No Applet
B2 C. Conjugates
B3 Pythagoras
B4 Distance
B5 Rt Triangle Similarity
B6 Trig., Functions
B7 Dot & Cross Products
B8 Cosine Law
B9 Exponential & cis fns
B10 Easy Trig Identities
B11 Set Viewpoint
Arithmetic Videos - Real Player Format
Decimal Addition Methods
Decimal
Subtraction Methods
Decimal
Multiplication Methods
Decimal Division Methods
Fractions
Primes
Greatest Common Divisor
Divisors
Least Common Multiples
Square Root
Simplification
Using formulas forwards
& Backwards - A unifying theme for algebra skill development - the 4th
skill in Volume 2, Three
Skills for Algebra!
What
is a Variable?
|