Its All about Pj Problem Strings (SiPjAjk) - 7 Spaces Of Interest (Si) and their associated Basic Sequences; 7 Pj Problems of Interest (PPI) and their Alleles (Ajk)

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) = x4/(x2 + x - 6)
(b) f(u) = (u -1)1/3
Ans: (a) Since x2 + x - 6 cannot be 0, function is not defined for values of x for which x2 + 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 - L2. 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 = x2 + 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 = x3. Surface area S = 5x2. Ratio V:S = f(x) = x3/5x2 = 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.



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