Categories

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forms

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.

CANthe 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.

Power Functions

A power function is any function that CAN be represented with a formula of the form

\[ pow(x) = k \, x^p \]

where $k$ and $p$ are real numbers.

Polynomial Functions

A polynomial function is any function that CAN be represented with a formula of the form

\[ poly(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_3 x^3 + a_2 x^2 + a_1 x^1 + a_0 x^0 \]

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.

Rational Functions

A rational function is any function that CAN be represented with a formula of the form

\[ rat(x) = \frac { a_n x^n + a_{n-1} x^{n-1} + \cdots + a_3 x^3 + a_2 x^2 + a_1 x + a_0 } { b_m x^m + b_{m-1} x^{m-1} + \cdots + b_3 x^3 + b_2 x^2 + b_1 x + b_0 } \]

where the $a_k$ and $b_k$ are real numbers and $a_n \ne 0$ and $b_m \ne 0$.

Radical/Root Functions

A radical or root function is any function that CAN be represented with a formula of the form

\[ rad(x) = A \sqrt [n]{B \, x + C} + D = A (B \, x + C)^{\tfrac {1}{n}} + D \]

where the $A$, $B$, $C$, and $D$ are real numbers and $A \ne 0$ and $B \ne 0$.

Exponential Functions

An exponential function is any function that CAN be represented with a formula of the form

\[ exp(x) = A \cdot r^{B \, x + C} \]

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.

Shifted Exponential Functions

A shifted exponential function is any function that CAN be represented with a formula of the form

\[ shexp(x) = A \cdot r^{B \, x + C} + D \]

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.

Logarithmic Functions

A logarithmic function is any function that CAN be represented with a formula of the form

\[ log(x) = A \log _r(B \, x + C) + D \]

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 $A \ln (B \, x + C) + D$.

Sine Functions

A sine function is any function that CAN be represented with a formula of the form

\[ sin(x) = A \sin (B \, x + C) + D \]

where $A$, $B$, $C$, and $D$ are real numbers.

Cosine Functions

A cosine function is any function that CAN be represented with a formula of the form

\[ cos(x) = A \cos (B \, x + C) + D \]

where $A$, $B$, $C$, and $D$ are real numbers.

Absolute Value Functions

An absolute value function is any function that CAN be represented with a formula of the form

\[ abs(x) = A | B \, x + C | + D \]

where $A$, $B$, $C$, and $D$ are real numbers.

Operations

The elementary function categories above 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 above 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 four operations, then our third choice is a composition. We study composition in great depth later.

There is a fourth choice and that is a piecewise defined function. Usually, these are easy to identify, because they have a big curly brace in front of several formulas.

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

2025-01-09 20:19:45