rigor
What does it mean to analyze a function?
Analyzing a function means listing its characteristics, features, and aspects along with an explanation of how you decided on these characteristics, features, and aspects.
We want BOTH:
A description of the characteristic
Your reasoning
Rigor lives inside your reasoning.
Rigor lives inside your explanations.
Rigor means that you are using proper and precise mathematical language to explain how you know you are correct. This includes explaining how you know you have accounted for everything.
The List
The list of function characteristics, features, and aspects doesn’t change.
For us, just starting out, the domain is a list of all real numbers that have been
paired with a function value (another real number).
The domain might be stated as part of the function definition. The domain
might be the natural domain implied by a formula. The domain might be the
collection of first coordinates from points on a graph. The domain might be
restricted by situational constraints. Your reasoning explains how you decided.
Domains are usually described with interval notation.
Discontinuities are domain numbers around which the function is behaving “wierd”.
For us, just starting out, “weird” means that close domain numbers do not have close
function values.
Singularities are non-domain numbers around which the function is behaving “wierd”.
In Algebra, functions are continuous over domain intervals. For us, just starting out, we generally identify discontinuities and singularities, remove those, and are left with domain intervals where the function is contniuous.
The end-behavior of a function is a simple description of how the function values behave as the domain values tend to or .
The behavior of a function is a simple description of how the function values change.
This includes where in the domain the function increases and decreases.
We will extend this to more detailed descriptions around discontinuities and singularities.
We will use rates of change to measure function behavior.
The extreme values of a function include the global maximum and global
minimum values. These are also called the absolute maximum and absolute
minimum values.
They also include local maximum and local minimum values. These are also
called relative maximum and relative minimum values.
Along with the maximum and minmum function values, we want to know where in the domain these occur.
The range is the collection of function values.
Domains are usually described with interval notation.
We want precise analysis. We want to be exact.
The only way to do this is by using algebra. Graphs are inherently inaccurate and
there is nothing we can do about this.
Our goal is rigorous algebraic reasong.
However, we are just starting out. We don’t have all of our algebraic tools yet. That
is what this course is supplying. So, sometimes we have to provide graphical
reasoning. It will be approximate rather than precise. We’ll just have to
understand this. When we can be precise with algebra, then that is what we want.
We want precise analysis, which comes through logical reasoning and algebraic
calculations, when we can get it.
We want to explain how we know we are exactly correct, but will settle for graphical
feelings of correct, if that is all we can get.
Learning when the algebra will not produce the reasoning we want is part of learning mathematics. Algebra and function reasoning first. Then graphical reasoning. But, always some reasoning. Always an explanation. Not just a declaration of facts.
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more examples can be found by following this link
More Examples of Visual Behavior