Study Notes for Calculus on Functions and Their Graphs

Function Definition: A function $f$ from a set $D$ (domain) to a set $Y$ is defined as a rule that assigns a unique value in $Y$ to each element $x$ in $D$.

A function assigns a unique element of set $Y$ to each element in set $D$.
Diagram Representation: Functions can be represented as machines that process inputs from the domain to produce outputs in the range.

Simple Functions and Domains:
1. $y = x^2$
  - Domain: $(- ext{infinity}, ext{infinity})$
  - Range: $[0, ext{infinity})$
2. $y = rac{1}{x}$
  - Domain: $(- ext{infinity}, 0) igcup (0, ext{infinity})$
  - Range: $(- ext{infinity}, 0) igcup (0, ext{infinity})$
3. $y = ext{sqrt}(x)$
  - Domain: $[0, ext{infinity})$
  - Range: $[0, ext{infinity})$
4. $y = ext{sqrt}(4-x)$
  - Domain: $(- ext{infinity}, 4)$
  - Range: $[0, ext{infinity})$
5. $y = ext{sqrt}(1-x^2)$
  - Domain: $[-1, 1]$
  - Range: $[0, 1]$

Calculating Domains:
- The domain of $y = x^2$ is $ ext{R}$, thus it gives a real y-value for any real number x. The range is $[0, ext{infinity})$ since squares of real numbers are non-negative.
- For $y = rac{1}{x}$, y is undefined at $x=0$, making the domain $(- ext{infinity}, 0) igcup (0, ext{infinity})$ with a range of $(- ext{infinity}, 0) igcup (0, ext{infinity})$.
- In $y = ext{sqrt}(x)$, the domain is $[0, ext{infinity})$ as square roots produce non-negative y-values.
- For $y = ext{sqrt}(4-x)$, x must satisfy $4-x
  eq 0$, hence x must be less than 4 which makes the domain $(- ext{infinity}, 4)$. The range is $[0, ext{infinity})$ since the results of the function are the non-negative square roots.
- The last example is $y = ext{sqrt}(1-x^2)$ with domain $[-1, 1]$ and range $[0, 1]$ as it represents a semicircle.

To determine the shape of a function's graph between points, we must plot points generated from the function and draw smooth curves.
A specific example includes making a table of x-y pairs satisfying equations, followed by plotting these pairs.

A smooth curve may approximate a function graphed based on paired data from tables.

Definition: A function $f$ can only output one value for each $x$ in its domain; thus a graph of a function must not be intersected by any vertical line more than once. This test determines if a curve in a coordinate plane represents a function. For example:
- A circle fails this test due to vertical lines intersecting it on two occasions while semicircles pass the test, indicating that they can represent functions.

Definition and Graphing: A piecewise-defined function applies different formulas across different segments of its domain.
Example: The function is defined as follows:
- $y = x$ when $x ext{ } | ext{ } x ext{ } orall x ext{ } | ext{ } 1$ when $x>1$. Thus, the function spans the entire set of real numbers even with varying formulas.
The Greatest Integer Function, denoted as $ ext{floor}(x)$ or $[x]$, yields the largest integer less than or equal to $x$. Its graph remains on or below the line $y=x$.
The Least Integer Function, denoted as $ ext{ceil}(x)$ or $ ext{ceil}(x)$, outputs the smallest integer greater than or equal to $x$, and it shows a graph above the line $y=x$.

Definitions: A function $f$ defined on an interval $I$ is labeled as:
1. Increasing: If $f(a) < f(b)$ for two distinct points $a$ and $b$ within $I$.
2. Decreasing: If $f(a) > f(b)$ for two distinct points $a$ and $b$ within $I$.

Definitions:
1. A function $f$ is even if $f(-x) = f(x)$ for all $x$ in the domain, which indicates symmetry about the y-axis.
2. A function $f$ is odd if $f(-x) = -f(x)$ for all $x$ in the domain, indicating symmetry about the origin.
Examples:
- An even function example is $f(x) = x^2$ which retains symmetry across the y-axis.
- An odd function example is $f(x) = x^3$ which maintains symmetry around the origin.

A linear function takes the form $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept, characterized as constant functions when $m = 0$.

Two variables $y$ and $x$ are proportional if there exists a nonzero constant $k$ such that $y = kx$.

A function defined as $f(x) = ax^n$, where $a$ is a constant and $n$ is a real number, is labeled a power function. Important cases are:
1. Positive Integer: Power functions defined for all real values of $x$ where $n>0$ produce continuous graphs.
2. Negative Integer: These functions have asymptotes as they approach discontinuities in their graphs.
3. Roots: The domain of the root function is defined for all $x$ in the real number system based on whether $n$ is even or odd.

Polynomial Definition: A polynomial $p$ consists of terms of the form $ax^n$, where $n$ is a non-negative integer and $a$ are real coefficients.
The domain of all polynomials is $ ext{R}$. The degree of a polynomial is the highest power of $x$.
Quadratic Functions: A second-degree polynomial $p(x) = ax^2 + bx + c$ is often referred to as quadratics.
Graphical Representation: Quadratic functions graph as parabolas, while cubic functions ($p(x) = ax^3 + bx^2 + cx + d$) show variable curves depending on their coefficients.

Definition: A rational function is the quotient of two polynomials, expressed as $R(x) = rac{p(x)}{q(x)}$; their graphs reveal asymptotic behavior where the function diverges.

Outcomes of algebraic operations involving polynomials (addition, subtraction, multiplication, division, and root extraction) fall under algebraic functions.

Various functions in trigonometry encompass sine, cosine, tangent, etc., also analyzed through graphical interpretations depicting periodic behavior and ranges.

A function represented as $f(x) = a^x$, where the base $a > 0$. The critical feature shows domain as $ ext{R}$ and the range as $(0, ext{infinity})$.

The definition states that if $y = a^x$, then $x = ext{log}_a(y)$; it expresses inverse relations to exponentials.

A function $f$ allows for an inverse $f^{-1}$ provided $f$ is a one-to-one function across its domain.
The method of graphing inverse functions entails reflecting graphs across the line $y=x$.

Functions, through their definitions, properties, and graphical interpretations, are foundational concepts in calculus discussed clearly, supplemented with comprehensive examples and theoretical implications for further exploration.