Functions and Models (Review) - Lecture Notes

This flashcard set covers the fundamental definitions, properties, and types of functions and models as discussed in Lecture Hand-out No. 1 and No. 2 by Engr. Oscar L. Poloyapoy.

Function

A rule of correspondence that assigns each element of a set $X$ to exactly one element of set $Y$.

Independent Variable

The variable $x$ representing the domain of the function; its values are chosen independently to determine the value of the outcome.

Dependent Variable

The variable $y$ representing the range of the function; its value depends upon the chosen value of $x$.

Domain

The set of all permissible or possible values of the independent variable ($x$).

Range

The set of all resulting values of the dependent variable ($y$).

Single-valued Function

A specific type of function where only one value of $y$ corresponds to each value of $x$ in the domain.

Notation $f: X \rightarrow Y$

Read as "$f$ takes set $X$ into set $Y$" or "$f$ is a function of $X$ into $Y$".

Relation

A correspondence that assigns each element of a set $X$ to at least one element of set $Y$; note that all functions are relations, but not all relations are functions.

Linear Function

First degree equations of the form $y = mx + b$ where $m$ and $b$ are constants, or $y = k$ where $k$ is a constant.

Rational Function

Functions that can be expressed in the form $h(x) = \frac{f(x)}{g(x)}$, where the domain is restricted to values where $g(x) \neq 0$.

Quadratic Function

A function represented by the equation $y = ax^{2} + bx + c$ where $a \neq 0$, resulting in a parabolic graph.

Composite Function

Written as $f(g(x))$ or $(f \text{ o } g)(x)$, it is the function obtained by replacing $x$ whenever it occurs in $f(x)$ by the expression for $g(x)$.

Even Function

A function where $f(-x) = f(x)$; its graph is symmetric with respect to the $y$-axis.

Odd Function

A function where $f(-x) = -f(x)$; its graph is symmetric with respect to the origin.

Inverse Function

Denoted as $f^{-1}$, it is a function with domain $Y$ and range $X$ such that $f^{-1}(y_{0}) = x_{0}$ if and only if $f(x_{0}) = y_{0}$.

Zeros of a Function

The values of $x$ for which $f(x) = 0$, corresponding to the $x$-intercepts of the graph.

Step-functions

Functions that jump from one value to the next, such as the piecewise rule used for first-class letter mailing costs.