Comprehensive Guide to Finding Domain and Range

Fundamental Definitions of Domain and Range

• Domain: The domain of a relation is defined as the set of all first coordinates. In a coordinate system, these correspond to the $x$-values of the relation.

• Range: The range of a relation is the set of all second coordinates. These correspond to the $y$-values or output values of the relation.

• Relationship and Notations: Identifying domain and range allows for understanding the limits of a given relation or function. They can be represented using discrete sets, interval notation, or set-builder notation.

Restrictions to Consider in Finding Domain and Range

When dealing with equations, certain mathematical conditions must be met to ensure values are defined within the set of real numbers:

• Radicals with Even Indices: For any radical expression where the index is even (e.g., square roots), the radicand (the value inside the radical) must be non-negative. This means it must be greater than or equal to zero ($\text{radicand} \geq 0$). Negative values under an even radical result in imaginary numbers, which are typically excluded in domain/range analysis unless specified.

• Fractions and Rational Expressions: In a rational function, the denominator must not be equal to zero ($\text{denominator} \neq 0$). Division by zero is undefined in mathematics; therefore, any value of $x$ or $y$ that results in a zero denominator must be excluded from the domain or range.

Domain and Range from Ordered Pairs

To find the domain and range from a set of ordered pairs, identify the unique first elements and the unique second elements.

• Example 1: Given the relation $F = \{ (1, 2), (2, 2), (3, 5), (4, 5) \}$

  • Domain: The first coordinates are $1$, $2$, $3$, and $4$. Thus, the domain is the set $\{ 1, 2, 3, 4 \}$.
  • Range: The second coordinates are $2$, $2$, $5$, and $5$. When listing elements of a set, duplicates are omitted. The range is the set $\{ 2, 5 \}$.

• Example 2: Given the relation with ordered pairs $(1, -1)$, $(2, -3)$, $(0, 5)$, $(-1, 3)$, $(4, -5)$, $(-1, 5)$, and $(4, -4)$.

  • Domain Calculation: Collecting all unique $x$-values: $1, 2, 0, -1, 4$. Arranged in ascending order, the domain is $\{ -1, 0, 1, 2, 4 \}$.
  • Range Calculation: Collecting all unique $y$-values: $-1, -3, 5, 3, -5, -4$. Arranged in ascending order (based on a number line), the range is $\{ -5, -4, -3, -1, 3, 5 \}$.

Domain and Range from Mapping Diagrams and Tables of Values

• Mapping Diagrams: In a mapping diagram, the input side (often on the left) represents the domain, and the output side (on the right) represents the range.

  • If input values are $\{ A, B, C, D \}$, then the Domain is $\{ A, B, C, D \}$.
  • If output values are $\{ X, Y, Z \}$, then the Range is $\{ X, Y, Z \}$.

• Table of Values (Example 1):

  • $x$-values: $-7$, $-4$, $2$, $8$, $10$, $19$. Domain = $\{ -7, -4, 2, 8, 10, 19 \}$.
  • $y$-values: $12$, $9$, $-11$, $17$, $6$, $-5$. Ascending order Range = $\{ -11, -5, 6, 9, 12, 17 \}$.

• Table of Values (Example 2):

  • $x$-values: $-19$, $-6$, $2$, $1$, $15$, $18$. Domain = $\{ -19, -6, 1, 2, 15, 18 \}$.
  • $y$-values: $-20$, $-3$, $4$, $8$, $9$, $16$. Range = $\{ -20, -3, 4, 8, 9, 16 \}$.

Domain and Range from Graphs

Graphs provide visual representations where domain is measured along the horizontal axis (left to right) and range is measured along the vertical axis (bottom to top).

• Finite Graph with Shaded Endpoints: If a graph starts with a shaded circle at $(-10, 3)$ and ends with a shaded circle at $(-2, 18)$:

  • Domain: The graph spans from $x = -10$ to $x = -2$. Using interval notation with brackets for included endpoints: $[ -10, -2 ]$. In set notation: $\{ x \mid -10 \leq x \leq -2 \}$.
  • Range: The graph spans from $y = 3$ up to $y = 18$. Range: $[ 3, 18 ]$. In set notation: $\{ y \mid 3 \leq y \leq 18 \}$.

• Graph with Open and Closed Circles: Consider a graph with an open circle at $(0, 1)$ and a shaded (closed) circle at $(4, -4)$.

  • Domain: Spans from $0$ (not included) to $4$ (included). Interval notation: $( 0, 4 ]$. Set notation: $\{ x \mid 0 < x \leq 4 \}$.
  • Range: The lowest point is $-4$ (shaded/included) and the highest point is $1$ (open/not included). Interval notation: $[ -4, 1 )$. Set notation: $\{ y \mid -4 \leq y < 1 \}$.

• Infinite Graphs (Arrows):

  • Full Line/Curve with Double Arrows: If a graph has arrows at both ends extending infinitely.
    • Domain: From negative infinity to positive infinity ($( -\infty, \infty )$). This is the set of all real numbers.
    • Range: Also the set of all real numbers ($y \in \mathbb{R}$).
  • Graph with Single Arrow (Ray-like): Consider a graph starting at an open circle near $(3, 16)$ and extending down toward the left with an arrow.
    • Domain: The $x$-values go from negative infinity up to but not including $3$. Solution: $x < 3$.
    • Range: The $y$-values go from negative infinity up to but not including $16$. Solution: $y < 16$.

Finding Domain and Range from Equations

To find the domain, solve the equation for $y$ in terms of $x$. To find the range, solve the equation for $x$ in terms of $y$.

• Case 1: Linear Polynomial ($2x - y = 5$):

  • Domain: Solve for $y$\rightarrow $-y = -2x + 5$\rightarrow $y = 2x - 5$. This is a polynomial with no restrictions. Domain: Set of all real numbers ($\mathbb{R}$).
  • Range: Solve for $x$\rightarrow $2x = y + 5$\rightarrow $x = \frac{y + 5}{2}$. This is also a polynomial structure without square roots or variables in denominators. Range: Set of all real numbers ($\mathbb{R}$).

• Case 2: Quadratic Equation ($y = x^2 - 9$):

  • Domain: $y$ is expressed as a polynomial. Domain: Set of all real numbers ($\mathbb{R}$).
  • Range: Solve for $x$\rightarrow $x^2 = y + 9$\rightarrow $x = \pm \sqrt{y + 9}$.
  • Apply restriction for radicals: $y + 9 \geq 0$\rightarrow $y \geq -9$.
  • Range: $\{ y \mid y \geq -9 \}$.

• Case 3: Another Quadratic Example ($x^2 + y - 4 = 0$):

  • Domain: Solve for $y$\rightarrow $y = 4 - x^2$. No restrictions. Domain: Set of all real numbers ($\mathbb{R}$).
  • Range: Solve for $x$\rightarrow $x^2 = 4 - y$\rightarrow $x = \pm \sqrt{4 - y}$.
  • Apply restriction: $4 - y \geq 0$\rightarrow $-y \geq -4$\rightarrow $y \leq 4$ (inequality symbol flips when multiplying by $-1$).
  • Range: $\{ y \mid y \leq 4 \}$.

• Case 4: Rational Equation with Multiple Variables ($3y + xy = 2x + 1$):

  • Domain: Group $y$ to solve in terms of $x$.
    • Factoring: $y(3 + x) = 2x + 1$.
    • Dividing: $y = \frac{2x + 1}{3 + x}$.
    • Restriction: Denominator $3 + x \neq 0$\rightarrow $x \neq -3$.
    • Domain: Set of all real numbers except $-3$.
  • Range: Group $x$ to solve in terms of $y$.
    • $3y + xy = 2x + 1$\rightarrow $xy - 2x = 1 - 3y$.
    • Factoring: $x(y - 2) = 1 - 3y$.
    • Dividing: $x = \frac{1 - 3y}{y - 2}$.
    • Restriction: Denominator $y - 2 \neq 0$\rightarrow $y \neq 2$.
    • Range: Set of all real numbers except $2$.