YOU are better than YOU think. Show yourself how:
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Read logic
chapters 1 to 5 in online volume Three
Skills for Algebra for greater skills & confidence
in work Read notes and textbooks like a lawyer, so that no nuance, no subtlety and no clause escapes your attention. |
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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;
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Caution: Site advice is approximately correct, for some circumstances, not all. That leaves room for thought |
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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.
Function Definition V.
For Math 536 - not for 436.
Inverse Functions -Reversing a Computation
The graph of the function y = g(x) = x2 looks like the following.
Here (x,y) is on the graph of g(x) = x2 when and only when y = x2.
For every real number x, there is a unique y, namely y = x2 such (x,y) belongs to the graph of g,
graph(g) = { [a,b] : b = a2 }
By construction, the vertical line rule holds
Now a horizontal line through a value y = c intersects the graph of at two, one or no points for c positive, zero and negative respectively.
First Domain Restriction Example
For the following graph of the computation rule h where h(x) = x2 for x nonnegative and h(x) is left undefined for x negative, the horizontal rule applies and it defines a function f.
Having drawn the graph of y = h(x) precisely, we can use the vertical line rule to compute h(x) = x2 and the horizontal line rule to compute f(x). See below.
So h maps a real number a to real number b with the property b = a2 while f maps the real number c to a real number d. Since h(d) = c, the number d has the property that d2 = c.
The diagram first suggests that applying h (the vertical line rule) to a number a > 0 gives a point b = h(a). The diagram then suggests f(b) = a. So the action of f undoes or inverts the action of h.
Likewise, the diagram first suggests that applying f (the horizontal line rule) to a number c > 0 gives a point d = f(c). The diagram then suggests h(d) = c. So the action of h undoes or inverts the action of f. .
In general, given a set of points S in the plane, if a pair of functions can be computed using the horizontal and vertical line rules with the set S, then each function inverts or undoes the other. They represent a pair of inverse functions. That may be discussed further in the topic Composition of Functions.
Remark 1. The natural logarithm ln(x) may be obtained as the inverse function for the (natural) exponential function exp(x) = ex, and vice-versa.
Remark 2. Inverses of trig functions (sine, cosine, tangent) and so on are obtain by domain restrictions that yield sets with the horizontal line properties. Which domain restriction to take may be a matter of convenience or convention. Read the manual for your calculator to determine how those inverses are defined.
Remark 3. With coordinates in the plane, we can describe or represent computation rules (functions) in standard and non-standard ways. The standard way puts the dependent variable first and independent variable second. Doing so gives the graph of the function f. The vertical line rule gives a means for finding the dependent variable y = f(x) from an the independent variable x. The non-standard way puts the dependent variable second and the independent variable first. Doing so provides a non-standard graph of the function x = h(y) - the standard graph reflected across the line y = x. That being said, the horizontal line rule gives a method for calculating x = h(y) from the independent variable y.
Remark 4. If a set of points in the plane, if the use of vertical and "horizontal" lines yields two different functions y =f(x) and x = h(y) between points on horizontal and vertical axes the pair of functions f and h are , inverse to each-other, with the range of one being domain of the other.
ADVERSTISEMENT
www.whyslopes.com
Analytic Geometry
Area Entrance
Entrance + Pages Below this page
Pages at Current LevelFNs & 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
Area Entrance (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
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.
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
SimplificationUsing formulas forwards & Backwards - A unifying theme for algebra skill development - the 4th skill in Volume 2, Three Skills for Algebra!