Categories

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Templates

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

Power Functions

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

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

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

Polynomial functions are sums of power functions with powers that are nonnegative integers (whole numbers).

Polynomial Functions

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

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

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

Rational Functions

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

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

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

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

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

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

where $A$, $B$, and $C$ are real numbers, $A$ is a nonzero real number, and $r$ is a positive real number.

Shifted Exponential Functions

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

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

where $A$, $B$, $C$, and $D$ are real numbers, $A \ne 0$ and $B \ne 0$, and $r$ is a positive real number.

Logarithmic Functions

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

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

where $A$, $B$, $C$, and $D$ are real numbers and $r > 0$.

The domain is all positive real numbers that make the inside positive.

Basic Sine Functions

A Basic Sine Function is any function that CAN be represented with a formula of the form

\[ S(x) = A \sin (x) \]

where $A$ is a real number.

Sine Functions

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

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

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

Basic Cosine Functions

A Basic Cosine Function is any function that CAN be represented with a formula of the form

\[ C(x) = A \cos (x) \]

where $A$ is a real number.

Cosine Functions

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

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

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

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

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

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more examples can be found by following this link
More Examples of Elementary Functions

2025-01-09 20:17:51