We are building a library of the elemntary functions. The idea is to use the library to
list characteristics, features, and aspects of all functions within each category.
That way, if we can identify the type of function we have, then we get free
information when analyzing functions.
The category becomes our reasoning.
These are “CAN” questions.
CAN the formula we are given be rewritten as one of the official standard forms for
each category?
Official Templates
These elementary function categories are our first choice. If a function can be
represented by one of these standard forms, then we want to describe the function
as one of these elementary functions. That gives us the most information.
A power function is any function that CAN be represented with a formula of the form
where \(k\) and \(p\) are real numbers.
A polynomial function is any function that CAN be represented with a formula of the form
where the \(a_k\) are real numbers and \(a_n \ne 0\).
Our order of preference inside the poynomial category is first constant function, then linear function, then quadratic function, then polynomial function.
A rational function is any function that CAN be represented with a formula of the form
where the \(a_k\) and \(b_k\) are real numbers and \(a_n \ne 0\) and \(b_m \ne 0\).
A radical or root function is any function that CAN be represented with a formula of the form
where the \(A\), \(B\), \(C\), and \(D\) are real numbers and \(A \ne 0\) and \(B \ne 0\).
An exponential function is any function that CAN be represented with a formula of the form
where \(A\), \(B\), and \(C\) are real numbers, \(A\) is a nonzero real number, and \(r\) is a positive real number.
We prefer \(e\) as the base.
A shifted exponential function is any function that CAN be represented with a formula of the form
where \(A\), \(B\), \(C\), and \(D\) are real numbers, \(A \ne 0\) and \(B \ne 0\), and \(r\) is a positive real number.
We prefer \(e\) as the base.
A logarithmic function is any function that CAN be represented with a formula of the form
where \(A\), \(B\), \(C\), and \(D\) are real numbers and \(r > 0\).
The domain is all positive real numbers that make the inside positive.
We prefer \(e\) as the base: \(\ln (x)\).
- If the base \(r = 10\), then we use the shorthand \(A \log (B \, x + C) + D\).
- If the base \(r = e\), then we use the shorthand \(A \ln (B \, x + C) + D\).
A sine function is any function that CAN be represented with a formula of the form
where \(A\), \(B\), \(C\), and \(D\) are real numbers.
A cosine function is any function that CAN be represented with a formula of the form
where \(A\), \(B\), \(C\), and \(D\) are real numbers.
An absolute value function is any function that CAN be represented with a formula of the form
where \(A\), \(B\), \(C\), and \(D\) are real numbers.
Operations
These elementary function categories are our first choice. If a function can be
represented by one of these standard forms, then we want to describe the function
as one of these elementary functions. That gives us the most information.
If a function cannot be identified as one of these elementary functions, then our
second choice is an operation.
These are “IS” questions.
IS the formula we are given written as one of the official standard forms for an
operation?
Our interpretation of every expression is through the Order of Operations.
Every mathematical expression involving mathematical operations is either
- a Sum;
- a Difference;
- a Product; or
- a Quotient;
Each mathematical expression is one of these and only one of these. The Order of Operations tells us which.
Composition
These elementary function categories are our first choice. If a function can be
represented by one of these standard forms, then we want to describe the function
as one of these elementary functions. That gives us the most information.
If a function cannot be identified as one of these elementary functions, then our
second choice is an operation.
If a function is not an elementary function and it is not one of our fouor operations,
then our third choice is a composition. We study composition is great depth later.
If a function is not any of these, then it is weird.
Incidentally, we like weird functions.
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more examples can be found by following this link
More Examples of Analysis