Functions, Domains, and Absolute Value Concepts

Fundamental Definition and Representation of Functions

• A function is defined as a specific type of relation between two sets.
• The core requirement of a function is that each input value maps to exactly one output value.
• Functions can be represented through various formats, including:
  • Data Points/Sets of Ordered Pairs: Groups of coordinates where the first value is the input and the second is the output. Example: $[(1, A), (2, B), (3, B), (4, C), (5, D)]$.
  • Graphs: Visual mappings of points on a coordinate plane.
  • Equations: Mathematical expressions defining the relationship between variables, such as $y = x^2$.

Identifying Functions and the Vertical Line Test

• Criteria for Set-Based Functions:
  • In a set of points like $\{ (1, 5), (2, 7), (3, 7), (4, 8) \}$, the relation is a function because every unique input is associated with exactly one output value (note that outputs can repeat, but inputs cannot).
  • In a set of points like $\{ (1, 5), (1, 7), (3, 10), (4, 15) \}$, the relation is NOT a function. This is because the input value $1$ repeats and is mapped to two different output values ($5$ and $7$).
• Criteria for Equations:
  • The equation $y = x^2$ represents a function.
  • The equation $x = y^2$ does NOT represent a function.
• Vertical Line Test:
  • This is a graphical method used to determine if an equation or a graph is a function.
  • If any vertical line intersects the graph more than once, the graph does not represent a function because a single $x$-value (input) would correspond to multiple $y$-values (outputs).

Algebraic Identification of Domain

• The domain of an algebraically defined function consists of all the real numbers for which the expression is defined. There are two primary restrictions to consider when finding the domain:
  1. Division by Zero: One cannot divide by zero. Any value that makes the denominator of a fraction equal to zero must be excluded from the domain.
  2. Even Roots of Negative Numbers: One cannot take an even root (square root, fourth root, etc.) of a negative number within the set of real numbers. The radicand (the expression under the radical) must be greater than or equal to zero.

Domain Calculation Examples

• Polynomial Functions:
  • For a function like $g(x) = 3x^2 + x + 3$, there are no denominators or radicals. Consequently, the domain is the set of all real numbers ($ℝ$), which is written in interval notation as $(-∞, ∞)$.
• Radical Functions:
  • Consider a function involving a square root, such as a function where we must ensure the radicand is non-negative.
  • Example: For an expression where the root requires $x - 5 ≥ 0$, we solve for $x$ to find $x ≥ 5$. The domain is then $[5, ∞)$.
• Rational Functions:
  • Example: $h(x) = \frac{x - 3}{x^2 - 16}$.
  • To find the domain, the denominator $x^2 - 16$ cannot be equal to zero ($x^2 - 16 ≠ 0$).
  • Solving for the excluded values: $x^2 ≠ 16$, which means $x ≠ ±\sqrt{16}$, so $x ≠ ±4$.
  • The domain is all real numbers except $-4$ and $4$. In interval notation, this is $(-∞, -4) ∪ (-4, 4) ∪ (4, ∞)$.

The Absolute Value Function

• The absolute value function is written as $f(x) = |x|$.
• Definition: The absolute value represents the distance of a number from zero, which is always a non-negative value.
  • If $x ≥ 0$, then $|x| = x$.
  • If $x < 0$, then $|x| = -x$ (which results in a positive value).
• Examples:
  • $|5| = 5$ because $5 ≥ 0$.
  • $|-3| = -(-3) = 3$ because $-3 < 0$.
• The Parent Graph: The graph of $y = |x|$ forms a characteristic "V" shape on the coordinate plane.