Core Functions

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In this course, we will begin studying a collection of functions called the Elementary Functions.

The Elementary Functions are created from an initial collection of “building block” functions, which we will call the Core Functions.

The Elementary Functions are created by combining the Core Functions together through addition, subtraction, multiplication, division, composition, and inverses.

The Elementary Functions are sums, differences, products, quotients, compositions, and inverses of Core Functions.

Calculus is largly about analyzing the Elementary Functions.

We will begin with a set of Core Functions:

Constant Functions: $C(x) = c$
The Identity Function: $Id(x) = x$
Power Functions: $P(x) = x^n$
Root Functions: $R(x) = \sqrt [n]{x}$
Exponential Functions: $E(x) = r^x$
Logarithmic Functions: $L(x) = \log _r(x)$
Sine and Cosine: $S(x) = \sin (x)$ and $C(x) = \cos (x)$
The Absolute Value Function: $AV(x) = |x|$

By forming sums, differences, products, quotients, compositions, and inverses of these Core Functions, we obtain the Elementary Functions.

Constant
Linear
Quadratic
Polynomial
Rational
Radical/Root
Exponential
Shifted Exponential
Logarithmic
Absolute Value
Trigonometric
Inverse Trigonometric

Creating the library of Elementray Functions is a long journey. We’ll need a lot of experience, which will take us right through Calculus.

The Core Functions

Core Constant Functions

A constant function is a function where all domain numbers are paired with the exact same range number. A constant function only has one value.

\[ Constant(x) = C \]

where $C$ is a fixed “constant” real number.

$\blacktriangleright \, H(t) = 3$

$\blacktriangleright \, H(t) = \sqrt {2}$

$\blacktriangleright \, H(t) = \pi $

$\blacktriangleright \, H(t) = \frac {3}{4}$

There is a core constant function for every real number.

The Identity Function

The is only one Identity function. The Identity function pairs every domain number with itself. The value of the Idenity function is just the domain value where it is evaluated.

\[ Id(x) = x \]

$\blacktriangleright \, Id(3) = 3$

$\blacktriangleright \, Id(\sqrt {2}) = \sqrt {2}$

$\blacktriangleright \, Id(\pi ) = \pi $

$\blacktriangleright \, Id \left ( \frac {3}{4} \right ) = \frac {3}{4}$

Core Power Functions

Core power functions raise the domain number to a power.

\[ Power(x) = x^r \]

where $r$ is a fixed non-zero real number.

Note: When $r$ is not an integer, then we need to pay close attention to the domain.

$\blacktriangleright \, Power(x) = x^3 $

$\blacktriangleright \, Power(x) = x^{-2} $

$\blacktriangleright \, Power(x) = x^{\sqrt {2}} $

$\blacktriangleright \, Power(x) = x^{\frac {2}{3}} $

There is a core power function for every real number.

Note: The power is non-zero, because $x^0 = 1$. That would just give us a constant function.

Core Root Functions

The value of a Core Root (Radical) function is a root of the domain number.

\[ Root(x) = \sqrt [n]{x} \]

where $n$ is a fixed positive real number.

These are often called “$n^{th}$” Root Functions.

Note: When $n$ is not a positive integer, then we need to pay close attention to the domain.

$\blacktriangleright \, Root(x) = \sqrt [2]{x} = \sqrt {x} $

$\blacktriangleright \, Root(x) = \sqrt [3]{x} $

$\blacktriangleright \, Root(x) = \sqrt [\tfrac {3}{4}]{x} $

$\blacktriangleright \, Root(x) = \sqrt [\pi ]{x} $

There is a core root or radical function for every real number.

Note: There is a bridge connecting power functions and root functions, which simplifies the algebra (later).

Core Exponential Functions

\[ r^x \]

where $r \ne 1$ is a positive real number.

Composition is a new operation for functions. By composing two functions, a new function is created.

Sometimes it is possible for the composition of two functions to be the Identity function, which is the identify element for function algebra.

This will be important as we build our library of Elementary Functions.

When the composition of two functions turns out to be the Identity function, then we say that the two component functions are “inverses” of each other.

The inverses of Exponential Functions are called Logarithmic function.

Core Logarithm Functions

\[ \log _r(x) \]

When we get to exponential and logarithmic functions, our magic sentence will be

$\log _r(a)$ is the thing you raise $r$ to, to get $a$.

There are plenty of relationships in our world that exhibit a periodic pattern.

Our Core Functions for periodic relationships will be Sine and Cosine.

Sine and Cosine

\[ \sin (x) \]

\[ \cos (x) \]

These will come from travelling around the unit circle.

Our last Core function is how we algebraically talk about distance. The Absolute Value function returns the distance between $0$ and the domain number.

The Absolute Value Function

\[ |x| = \begin{cases} -x & \text {if $x<0$,} \\ x & \text {if $x\ge 0$} \end{cases} \]

We can describe the distance between any two number on the number line using the absolute value function.

Suppose $a$ and $b$ are two real numbers on the real number line.

\[ |a-b| = \begin{cases} -(a-b) & \text {if $a<b$,} \\ a-b & \text {if $b\ge a$} \end{cases} \]

$|a-b| = |b-a|$

Both are positive or $0$.

Either describes the distance between $a$ and $b$.

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2026-06-13 18:07:50