Chapter 1: Functions and their Graphs

Today in class, 12/11/2015, we discussed how to make sense of technical writing.

Technical writing is the kind of writings in textbooks. When mathematicians write, they write in the most concise way possible. Writing with concision allows mathematicians to express exactly what they mean without any room for interpretation. Technical writing in math textbooks are written with every word chosen very carefully for its meaning.

In essence, mathematical writing is written to be as short and meaning packed as possible.

In an effort to shorten writing, however, mathematicians often include symbols, visuals, or shortened notation. For example, the graph below could be provided in lieu of an explanation, or the blue line that runs vertically through a point might be represented by the function x = 3⅙.

To understand this mathematical writing, one must understand the notation. A major downfall to shortening the ideas provided is a possible loss of comprehension on the part of the consumer. Therefore, in order to understand mathematical writing, one must learn the terms, symbols, and notation. One of the most important part to this is learning definitions.

In math, definitions are the shortened down explanations and notations that mathematicians use. Being able to process and comprehend mathematical definitions is a major part of reading the technical writing in math textbooks. 

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Definitions:

A function ƒ is increasing on an interval if, for any X₁ and X₂ in the interval,

X₁ < X₂ implies ƒ(X₁) < ƒ(X₂).

This can be read more simply. For any two points on an interval, With the first _________one on the left of the second, the second one is greater, and the line increases. _____________________________________________________________________________

A function ƒ is decreasing on an interval if, for any X₁ and X₂ in the interval,

X₁ < X₂ implies ƒ(X₁) > ƒ(X₂).

Same for this one as well. For any two points on an interval, With the first _________one on the left of the second, the first one is greater, and the line decreases.

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A function value ƒ(a) is called a relative maximum of ƒ if there exists an interval ___(X₁, X₂) that contains a such that

X₁ < X < X₂ implies ƒ(a) ≥ ƒ(X).

A Y coordinate of a function exist between two points that is greater than all  _______other points of that interval.

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A function value ƒ(a) is called a relative maximum of ƒ if there exists an interval ___(X₁, X₂) that contains a such that

X₁ X < X₂ implies ƒ(a) ≤ ƒ(X).

A Y coordinate of a function exist between two points that is less than all  _______other points of that interval.

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A function ƒ is even if, for each X in the domain of ƒ, 

ƒ(-X) = ƒ(X)

For each point Y, the X coordinate and its corresponding -X coordinate will be _________the same, and the graph with be symmetrical about the Y-axis.

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A function ƒ is odd if, for each X in the domain of ƒ, 

ƒ(-X) = -ƒ(X)

For each point (X,Y), its corresponding -X coordinate will be at -Y, and the graph _______with be symmetrical about the origin.