Domain and Range - Algebra-1

1. Fundamental Concepts

Domain: Refers to the set of all possible input values (independent variable, usually x) in a function, that is, the range of values the independent variable can take for the function to be meaningful. For example, in the function \(y = 2x\), the domain is all real numbers (because any real number substituted for x can yield a corresponding y).
Range: Refers to the set of all possible output values (dependent variable, usually y) in a function, that is, the range of all results the dependent variable can obtain when the independent variable takes all values within the domain. For example, in the function \(y = 2x\), the range is also all real numbers (because when x is any real number, y can also take any real number).

Essential Differences:
- The domain focuses on "the possibility of inputs" and answers "what values the independent variable can take";
- The range focuses on "the possibility of outputs" and answers "what values the dependent variable can obtain".
Dependence Relationship: The range is determined by both the domain and the function relationship. That is, the domain comes first, and the range can be determined through function operations. The domain is the restriction of the function itself on the independent variable (such as avoiding a denominator of 0, ensuring the number under a square root is non-negative, etc.).
Representation Methods: Both can be expressed using sets, inequalities, or intervals. For example, "all non-negative real numbers" can be expressed as \(\{x \in \mathbb{R} | x \geq 0\}\) (for the domain) or \(\{y \in \mathbb{R} | y \geq 0\}\) (for the range).

Example 1: Linear function \(y = 2x\)
- Domain: x can take all real numbers, i.e., \(D: \mathbb{R}\) (all real numbers).
- Range: Since x is all real numbers, the result of \(y = 2x\) is also all real numbers, i.e., \(R: \mathbb{R}\).
- Distinction: The domain is the range of x values, and the range is the range of y values. Although their ranges are the same here, they essentially differ as input and output.
Example 2: Quadratic function \(y = x^2\)
- Domain: x can take all real numbers (no restrictions), i.e., \(D: \mathbb{R}\).
- Range: Since the square of any real number is non-negative, the result of y is all non-negative real numbers, i.e., \(R: \{y \in \mathbb{R} | y \geq 0\}\).
- Distinction: The domain is " x can take any real number", and the range is " y can only take non-negative numbers", reflecting the difference between the input and output ranges.
Example 3: Square root function \(y = \sqrt{x}\)
- Domain: The number under the square root must be non-negative, so \(x \geq 0\), i.e., \(D: \{x \in \mathbb{R} | x \geq 0\}\).
- Range: The result of a square root is non-negative, so \(y \geq 0\), i.e., \(R: \{y \in \mathbb{R} | y \geq 0\}\).
- Distinction: Here, the domain and range have the same range, but the domain is a restriction on x (x cannot be negative), and the range is a characteristic of the result of y (y is non-negative). Essentially, they still differ as input and output.

Methods to Identify the Domain:
1. Start from the function expression and find the values of the independent variable that make the function meaningless (such as a denominator of 0, a negative number under a square root, a non-positive number as the argument of a logarithm, etc.);
2. Exclude these meaningless values, and the remaining part is the domain. For example, in the function \(y=\frac{1}{x}\), \(x = 0\) will make the denominator 0, so the domain is all real numbers except \(x = 0\).
Methods to Identify the Range:
1. First determine the domain;
2. Analyze the range of changes of the dependent variable when the independent variable takes values within the domain, based on the properties of the function (such as monotonicity, parity, image characteristics, etc.);
3. Derive the range. For example, for the function \(y=x^2 + 1\), the domain is \(\mathbb{R}\). Since \(x^2\geq0\), then \(y=x^2 + 1\geq1\), so the range is \(y\geq1\).
Quick Distinction Skills: Remember that "the domain is the range of x values, and the range is the range of y values". When analyzing, clarify "what is the input and what is the output", and judge based on the roles of the independent variable and the dependent variable in the function expression.