Introduction to Algebraic Notations, Domain, Range, and Intercepts

Inequality Notation: This is the most basic form of notation. It uses symbols such as less than ( < ), greater than ( > ), and less than or equal to ( $\le$ ). These symbols are used to compare a variable ( $m$ or $n$ ) to its left and right boundaries.
Interval Notation: This notation describes only the two boundaries of a set and does not utilize variables (no letters). For example, a set bounded by $-5$ on the left and $0$ on the right is represented using these values and specific boundary markers.
Set-Builder Notation: This notation includes both a variable and a description of the elements within the set. It often utilizes the "such that" symbol, which is a vertical bar ( $|$ ).
- Example: \{x | -5 < x \le 0 \} is read as "the variable $x$ such that $x$ is greater than $-5$ and less than or equal to $0$ ."
- Implicit Assumptions: When using inequality or set-builder notation, it is generally assumed that the set includes all real numbers ( $\text{reals}$ ) such as $0.01$ or $0.001$ unless specified otherwise (e.g., restricted to integers).

Open Boundaries (Not Included): When a boundary value is not part of the set, it is represented by an open circle on a number line and a parenthesis ( $($ or $)$ ) in interval notation. Example: a boundary at $-3.5$ might have an open circle.
Closed Boundaries (Included): When a boundary value is included in the set, it is represented by a shaded (low) circle on a number line and a bracket ( $[$ or $]$ ) in interval notation. Example: a boundary at $0$ that is shaded.
Infinity Boundaries: Infinity ( $\infty$ ) and negative infinity ( $-\infty$ ) are always coupled with parentheses because they can never be reached or bounded; they are always open.

Domain: Refers to all possible $x$ -values or inputs of a function. On a coordinate plane, the domain is determined by observing the graph from left to right.
Range: Refers to all possible $y$ -values or outputs of a function. On a coordinate plane, the range is determined by observing the graph from the bottom (down) to the top (up).

Linear Functions: These functions typically extend infinitely in both directions. Their domain and range are both $(-\infty, \infty)$ .
Quadratic Functions (Parabolas):
- Domain: Typically $(-\infty, \infty)$ as the graph continues left and right forever.
- Range: Dependent on the vertex. If the parabola opens upward from the origin ( $x$ -intercept), the range might be $[0, \infty)$ . If it has a maximum point at $y = 4.5$ , the range would be $(-\infty, 4.5]$ .
Constant Functions: These represent horizontal lines.
- Domain: $(-\infty, \infty)$ .
- Range: A single, specific $y$ -value that never changes.
Discontinuous Graphs: If a graph has a break (e.g., at $x = 1$ ), the domain must reflect that gap. For a graph with a break at $1$ , the domain might be expressed as (-\infty < x < 1) and (1 < x < \infty).

X-Intercept: The point where the graph crosses the $x$ -axis. At this point, the $y$ -value is always zero ( $y = 0$ ). This is also known as the "zero of the function." To solve for it algebraically, substitute $0$ for $y$ in the equation and solve for $x$ .
Y-Intercept: The point where the graph crosses the $y$ -axis. At this point, the $x$ -value is always zero ( $x = 0$ ). To solve for it, substitute $0$ for $x$ in the equation and solve for $y$ .

Equation: $y = 900 - 2x$ .
- $y$ represents the altitude (height in feet).
- $x$ represents time in seconds.
Y-Intercept Calculation: Setting $x = 0$ gives $y = 900$ .
- Interpretation: At initial time ( $0$ seconds), the balloon's maximum height is $900$ feet.
X-Intercept Calculation: Setting $y = 0$ results in $0 = 900 - 2x$ , which simplifies to $2x = 900$ , or $x = 450$ .
- Interpretation: The balloon reaches the ground ( $0$ feet) after $450$ seconds.

Question: I don't know if I'm gonna go with this step-builder or interval. I'm kinda confused on that part.
Response: For interval notation, use the boundaries. The minimum value for example one is $-2$ . Because it is less than or equal to, it is closed, requiring a bracket. The upper bound is $3$ , which is open, requiring a parenthesis. The interval is $[-2, 3)$ .
Question: What about the set-builder? These squiggly lines look like fleas based on how I draw them.
Response: The squiggly lines are braces. In set-builder notation, you follow a formula: the variable (e.g., $x$ ), then the "such that" bar, then the description. For example, $x$ such that -2 \le x < 3. If no further description is provided, it is assumed $x$ is a real number.
Question: Can x go past 100? If we have the lower bound $x \ge 4$ , where does it stop?
Response: It doesn't stop at $100$ ; it goes to infinity. We write this as x < \infty. Since we never reach infinity, it always gets a parenthesis: $[4, \infty)$ .