Domain and Range of Functions
Domain and range of functions is a Grade 11 foundational concept in Big Ideas Math. The domain is the complete set of valid input values (x-values) for which the function is defined. The range is the complete set of output values (y-values) the function produces. For f(x) = √(x − 3), the domain requires x − 3 ≥ 0, giving domain [3, ∞). For f(x) = 1/(x − 2), x ≠ 2, giving domain (−∞, 2) ∪ (2, ∞). Range is found by analyzing what output values are possible. Understanding domain and range is essential for function composition, inverse functions, and real-world modeling with restricted variables.
Key Concepts
For any function $f$, the domain is the set of all possible input values ($x$ values) for which the function is defined. The range is the set of all possible output values ($y$ values) that the function can produce.
Common Questions
What is the domain of a function?
The domain is the set of all valid x-values (inputs) for which the function is defined. It includes all x-values that produce a real output.
What is the range of a function?
The range is the set of all y-values (outputs) the function can produce for the inputs in its domain.
How do you find the domain of f(x) = √(x − 3)?
The expression under a square root must be ≥ 0: x − 3 ≥ 0, so x ≥ 3. Domain: [3, ∞).
How do you find the domain of f(x) = 1/(x − 2)?
Division by zero is undefined, so x − 2 ≠ 0, meaning x ≠ 2. Domain: (−∞, 2) ∪ (2, ∞).
How do you find the range of a function?
Analyze what output values are achievable: consider the function's behavior (minimum/maximum values, horizontal asymptotes), graph it, or solve y = f(x) for x and find which y-values allow real x-solutions.
How is domain written in interval notation?
Use square brackets [ ] for included endpoints and parentheses ( ) for excluded endpoints or infinity. For example, [3, ∞) means x ≥ 3 (3 included, ∞ always uses parenthesis).