*Its All about Pj Problem Strings (S _{i}P_{j}A_{jk}) -
7 Spaces Of Interest (S_{i}) and their associated Basic Sequences; 7 Pj Problems of Interest (PPI) and their Alleles (A_{jk})*

Expressions Of Pj Problems.

Pj Problems - Overview

Celestial Stars As Expressions Of Pj Problems

The Number Line As Expression Of Pj Problems

Geometries As Expressions Of Pj Problems

7 Spaces Of Interest - Overview

Triadic Unit Mesh

Creation As Expression Of Pj Problems

The Atom As Expression Of Pj Problems

Survival As Expression Of Pj Problems

Energy As Expression Of Pj Problems

Light As Expression Of Pj Problems

Heat As Expression Of Pj Problems

Sound As Expression Of Pj Problems

Music As Expression Of Pj Problems

Language As Expression of Pj Problems

Stories As Expressions of Pj Problems

Work As Expression Of Pj Problems

States Of Matter As Expressions Of Pj Problems

Buoyancy As Expression Of Pj Problems

Nuclear Reactions As Expressions Of Pj Problems

Molecular Shapes As Expressions Of Pj Problems

Electron Configurations As Expressions Of Pj Problems

Chemical Bonds As Expressions Of Pj Problems

Energy Conversion As Expression Of Pj Problems

Chemical Reactions As Expressions Of Pj Problems

Electromagnetism As Expression Of Pj Problems

Continuity As Expression Of Pj Problems

Growth As Expression Of Pj Problems

Human-cells As Expressions Of Pj Problems

Proteins As Expressions Of Pj Problems

Nucleic Acids As Expressions Of Pj Problems

COHN - Nature's Engineering Of The Human Body

The Human-Body Systems As Expressions Of Pj Problems

Vision As Expression Of Pj Problems

Walking As Expression Of Pj Problems

Behaviors As Expressions Of Pj Problems

Sensors' Sensings As Expressions Of Pj Problems

Beauty As Expression Of Pj Problems

Faith, Love, Charity As Expressions Of Pj Problems

Photosynthesis As Expressions Of Pj Problems

Weather As Expression Of Pj Problems

Systems As Expressions Of Pj Problems

Algorithms As Expressions Of Pj Problems

Tools As Expressions Of Pj Problems

Networks As Expressions Of Pj Problems

Search As Expressions Of Pj Problems

Differential Calculus As Expression Of Pj Problems

Antiderivative As Expression Of Pj Problems

Integral Calculus As Expression Of Pj Problems

Economies As Expresions Of Pj Problems

Inflation As Expression Of Pj Problems

Markets As Expressions Of Pj Problems

Money Supply As Expression Of Pj Problems

Painting As Expressions Of Pj Problems

Single Variable Functions - Domains

A functions *f*, is a *one-one mapping* from its *domain* to its range. A single variable function *f* is often expressed in association with its independent and dependent variables as:

*f(x) = y = an expression of x* (other letters can be used). In this representation, *x* is the *independent variable* (it constitutes the domain of *f*) and *y* is the dependent variable (it constitutes the range of *f*

1. What is meant by:

(a) The *domain* of a *function*?

(b) The *range* of a *function*?
**Ans:**1(a) The values that can be assigned to the independent variable of a function.

1(b) The values of the dependent variable evaluated at each value of the independent variable. For example, f(x) at x = 2 is the value f(2) and is a value in the range of the function *f*.

2. A curve has infinitely many points at x = -1. Is this curve the graph of a function?
**Ans:** No. There must be a one-one correspondence between the domain and range of a function.

3. Determine the domains of the following functions:

(a) f(x) = x^{4}/(x^{2} + x - 6)

(b) f(u) = (u -1)^{1/3}
**Ans:** (a) Since x^{2} + x - 6 cannot be 0, function is not defined for values of x for which x^{2} + x - 6 = (x + 3)(x - 2) = 0; i.e. for x= -3 or 2. Therefore domain of function is {all x ∈ R | x ≠ -3; 2}. Where R = all real numbers.

(b) Function is defined for every u, since every real number has a cube root. So the domain is {all u ∈ R}.

4. What part of the real number line is excluded from the domain of the following function:

f(x) = x/|x|
**Ans:** f(x) = x/x = 1 for x > 0; f(x) = x/-x = -1 for x< 0. undefined for x = 0.

So the domain is {all x ∈ R | x ≠ 0;}. So x = 0 is excluded.

5. What is the least number in the domain of the following function:

f(x) = 3 - 2x
**Ans:** The graph of the function intercepts the y-axis at 3 and the x- axis at 3/2, then it tends to infinity at both ends. Domain is {all x ∈ R}. The least number in this domain is -∞.

6. What part of the domain of the following function refers to the horizontal line in the graph of the following function:

G(x) = |x| + x
**Ans:** Function evaluates to G(x) = 2x; x ≥ 0; G(x) = 0; x < 0. Domain is {all x ∈ R}. The part of the graph for x < 0 is horizontal.

7. The perimeter of a rectangular area is 20.

(a) What is the domain of the area when expressed as a function of its length.

(b) What is the domain of the area when expressed as a function of its length and the length is restricted to be larger than the width.
**Ans:** (a) Let the width = W and the length = L. Then 2W + 2L = 20

So, W = (20 - 2L)/2. Then Area = [(20 - 2L)/2]L = 10L - L^{2}. Assuming positive length. Domain of area is 0 < L < 10.

Restriction requires L > W. So L > (20 - 2L)/2. Therefore domain of area is 5 < L < 10.

8. The volume of a cubic box is 2. The surface area is S = x^{2} + 4xh (where h is the height of the box). What is the domain of the function that expresses the ratio of the volume of the box to its surface area.
**Ans:** Let a side of the box be x. Then Volume of cubic box, V = 2 = x^{3}. Surface area S = 5x^{2}. Ratio V:S = f(x) = x^{3}/5x^{2} = x/5. Domain is x > 0.

9. Given that f(x) is an even function and the point (5,3) is on its graph. Indicate a point that must also be on the graph.
**Ans:** An even function is symmetric with respect to the y-axis. Since the point (5,3) is on the graph, the point (-5,3) must also be on the graph.

10. Given that f(x) is an odd function and the point (5,3) is on the graph. Indicate a point that must also be on the graph.
**Ans:** An odd function is symmetric with respect to the origin. Since the point (5,3) is on the graph, the point (-5,-3) must also be on the graph.

Mind Warm Ups

The *point* "**.**" is a mathematical abstraction. It has negligible size and a great sense of position. Consequently, it is front and center in abstract existential reasoning.

*Problems by Peter O. Sagay*