Functions, Graph Characteristics, and Two-Variable Linear Inequalities

Core Definitions of Functions, Domain, and Range

• Function:

  • A mathematical rule that assigns to each input exactly one output.
  • Associates an input variable ($x$) from the domain to a unique output variable ($y$) in the range.

• Domain:

  • The set of all allowable input ($x$) values of a function.
  • For all linear functions, the domain is the set of all real numbers ($(-\infty, \infty)$), unless the domain is explicitly restricted.

• Range:

  • The set of all possible output ($y$) values resulting from evaluating the domain through the function rule.

Linear Functions, Slope, and Rate of Change

• Linear Function Definition and Degree:

  • The highest degree of a linear function is $1$.
  • Example linear function: $f(x) = 2x + 3$.
  • Linear parent function: $y = x$.
  • Graphically, a linear function forms a continuous straight line with arrows on both ends, indicating indefinite extension in both directions.

• Slope-Intercept Form and Components:

  • Standard slope-intercept form is given by $y = mx + b$.
  • $m$ represents the slope of the line.
  • $b$ represents the $y$-intercept.

• Slope and Rate of Change:

  • Slope represents the constant rate of change of a line: $\text{Constant rate of change} = \text{slope}$.
  • Calculated between two points using the formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$.
  • Applies across discrete sets (distinct, separate points) and continuous contexts (unbroken intervals).

• Algebraic Expressions and Function Evaluation:

  • Consecutive quantities or expressions can be modeled algebraically, such as $2n + 2(n + 1)$.
  • Evaluating a linear rule for a given input value:
  • Given the expression $f(x) = 2x + 7$ evaluated at $x = -3$:
  • $f(-3) = 2(-3) + 7$
  • $f(-3) = -6 + 7 = 1$

Function Families: Quadratic, Constant, and Absolute Value

• Quadratic Functions:

  • The highest degree of a quadratic function is $2$.
  • Example quadratic function: $f(x) = x^2 - 2x + 1$.
  • Quadratic parent function: $y = x^2$.
  • Domain of Quadratic Parent Function: Expressed in interval notation as $(-\infty, \infty)$.
  • Range of Quadratic Parent Function:
  • Interval notation: $[0, \infty)$
  • Set-builder notation: $R: \{y \mid y \ge 0\}$

• Constant Functions:

  • Functions producing a horizontal line across all real inputs.
  • Domain: $(-\infty, \infty)$.
  • Range: Contains only the single constant output value (for example, Range: $\{5\}$).

• Absolute Value Function:

  • Standard parent equation: $f(x) = |x|$.
  • Forms a characteristic V-shaped graph with a vertex at the origin and rays extending outward indefinitely in both directions.
  • 

Five Key Graph Characteristics

• 1. $y$-Intercept:

  • The coordinate point where the graph crosses or intersects the vertical $y$-axis.
  • Written in coordinate notation as $(0, y)$.

• 2. $x$-Intercept:

  • The coordinate point where the graph crosses or intersects the horizontal $x$-axis.
  • Written in coordinate notation as $(x, 0)$.

• 3. Increasing and Decreasing Intervals:

  • Analysis of the behavior of $y$-values as $x$-values increase from left to right across the domain.
  • Increasing intervals: regions where the graph rises from left to right.
  • Decreasing intervals: regions where the graph falls from left to right.

• 4. Domain:

  • The complete set of all possible horizontal input ($x$) values covered by the graph.

• 5. Range:

  • The complete set of all possible vertical output ($y$) values achieved by the graph.

Two-Variable Linear Inequalities and Graphing Rules

• Inequality Symbols and Verbal Translations:

  • Less than: $<$
  • Example: $y < x + 3$
  • Greater than: $>$
  • Example: $y > -3x + 5$
  • Less than or equal to: $\le$
  • Example: $y \le 2x - 4$
  • Greater than or equal to: $\ge$
  • Example: $y \ge 5x + 7$

• Boundary Line Types:

  • 
  • Solid Line:
  • Used for inclusive inequalities ($\ge$ and $\le$).
  • Points lying directly on the solid boundary line are included in the solution set.
  • Dashed or Dotted Line:
  • Used for strict inequalities ($>$ and $<$).
  • Points lying directly on the dashed boundary line are NOT included in the solution set.

• Shading Rules for Solution Regions:

  • Greater than symbols ($>$ or $\ge$) require shading above the boundary line.
  • Less than symbols ($<$ or $\le$) require shading below the boundary line.

• Inequality Sign Inversion Rule:

  • When multiplying or dividing both sides of an inequality by a negative number, you must switch (reverse) the direction of the inequality sign.