HB 1.3 Recognizing Functions and Function Families

Relation: A relation is defined as the mapping between a set of input values, known as the domain, and a set of output values, known as the range.
Domain: The set of all possible input values for a relation.
Range: The set of all possible output values for a relation.
Function: A function is a specific type of relation that assigns to each element of the domain exactly one element of the range. While all functions are categorized as relations, only some relations qualify as functions.

Relations can be represented through various formats, including:

Ordered Pairs: A set of coordinates, such as $\{(-2, 2), (0, 2), (3, -4), (3, 5)\}$ .
Equations: Algebraic expressions like $y = \frac{2}{3}x - 1$ .
Verbal Statements: Descriptions of relationships, such as the relation between students in a school and each student's birthday.
Graphs: Visual plots on a coordinate plane.
Mappings: Diagrams showing arrows from domain elements to range elements.
Tables: Organized columns or rows listing domain and range values.

Function Notation: This is a method of representing functions algebraically. The notation $f(x)$ is read as "f of x." It indicates that $x$ is the independent variable. If $f$ is a function and $x$ is an element of its domain, then $f(x)$ denotes the output of $f$ corresponding to the input $x$ .
Scenario Modeling: Consider a T-shirt company that charges $15$ dollars for a one-time design setup plus $8$ dollars per shirt.
- The equation for the total cost is $y = 8x + 15$ .
- In function notation, this is written as $f(x) = 8x + 15$ .
- Alternatively, it can be written as $C(s) = 8s + 15$ , where cost $C$ is a function of the number of shirts ordered, $s$ .
- The independent variable $x$ or $s$ represents the shirts ordered (domain), and the dependent variable $y$ or $f(x)$ represents the total cost (range).

Domain and range can be expressed using set notation in specific real-world contexts:

Pencil Supply Example: A teacher gives $12$ pencils to every new student in a tutoring group. He expects no more than $6$ new students.
- The domain is list of possible student counts: $\{0, 1, 2, 3, 4, 5, 6\}$ .
- The range is the possible total pencils needed: $\{0, 12, 24, 36, 48, 60, 72\}$ .
Bakery Example: A neighborhood bakery sells bagels in sets of $6$ . Only the first $9$ customers can purchase them.
- The domain represents the number of customers: $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$ .
- The range represents the total number of bagels sold: $\{6, 12, 18, 24, 30, 36, 42, 48, 54\}$ .
Inequality Notation for Continuous Graphs:
- Example $g(x)$ : The domain is all real numbers greater than $-2$ and less than or equal to $5$ , written as -2 < x \le 5. The range is all real numbers greater than or equal to $-8$ and less than or equal to $8$ , written as $-8 \le y \le 8$ .
- Example $h(x)$ : The domain is the set of all real numbers, written as -\infty < x < \infty. The range is all real numbers less than or equal to $8$ , written as $y \le 8$ .

Vertical Line Test: This is a visual method used to determine if a relation on a graph is a function. If any vertical line drawn on the graph intersects the relation at more than one point, the relation is not a function.
Discrete Graph: A graph consisting of isolated, distinct points.
Continuous Graph: A graph of points connected by a line or smooth curve, containing no breaks.

Graphical behavior is determined by reading a graph from left to right:

Increasing Function: A function where the value of the dependent variable increases as the value of the independent variable increases across the entire domain.
Decreasing Function: A function where the value of the dependent variable decreases as the value of the independent variable increases across the entire domain.
Constant Function: A function where the value of the dependent variable remains the same over the entire domain. This occurs in linear functions $f(x) = ax + b$ when $a = 0$ , resulting in $f(x) = b$ .
Combination: Some functions exhibit different behaviors (increasing and decreasing) on different parts of their domain.

Functions are grouped into families based on shared characteristics and algebraic forms:

Linear Functions: These functions include those in the form $f(x) = ax + b$ , where $a$ and $b$ are real numbers. Their graphs are straight lines.
Exponential Functions: These functions include those in the form $f(x) = a \cdot b^x$ , where $a$ and $b$ are real numbers, and b > 0 but $b \neq 1$ . Their graphs represent rapid growth or decay.
Quadratic Functions: These functions include those in the form $f(x) = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are real numbers, and $a \neq 0$ . These functions often exhibit a "U-shape" (parabola) and contain either an absolute maximum or absolute minimum.

Absolute Minimum: A function has an absolute minimum if there is a point on the graph with a $y$ -coordinate that is less than the $y$ -coordinate of every other point on the graph.
Absolute Maximum: A function has an absolute maximum if there is a point on the graph with a $y$ -coordinate that is greater than the $y$ -coordinate of every other point on the graph.
x-intercept: The point where a graph crosses the $x$ -axis ( $y = 0$ ).
y-intercept: The point where a graph crosses the $y$ -axis ( $x = 0$ ).