Overview
This lecture covers the definition of the domain of functions, strategies for finding the domain, and provides specific examples of determining domains using interval notation.
Definition of Domain
- The domain is the set of all real numbers x where f(x) is also a real number.
- The domain is a list of all input values for which the function produces real outputs.
- In real-life contexts, inputs that do not make sense (like negative lengths) are excluded from the domain.
Steps for Finding the Domain
- Start with all real numbers as potential inputs.
- Remove values that make any denominator equal to zero.
- Remove values that create a negative number under an even root (e.g., square roots).
- Exclude values that do not fit real-world context or application problems.
- Express the solution in interval notation.
Example 1: Rational Function
- For ( f(x) = \frac{x^2 + 5}{x + 2} ), the denominator is zero when ( x = -2 ).
- The domain is all real numbers except ( x = -2 ).
- Interval notation: ( (-\infty, -2) \cup (-2, \infty) ).
Example 2: Quadratic Denominator
- For ( f(t) = \frac{t + 1}{t^2 - t - 2} ), set the denominator to zero: ( t^2 - t - 2 = 0 ).
- Factor to find ( t = 2 ) and ( t = -1 ).
- The domain excludes ( t = 2 ) and ( t = -1 ).
- Interval notation: ( (-\infty, -1) \cup (-1, 2) \cup (2, \infty) ).
Example 3: Square Root Function
- For ( f(x) = \sqrt{2x + 6} ), require ( 2x + 6 \geq 0 ).
- Solve to get ( x \geq -3 ).
- The domain is ( [-3, \infty) ).
Key Terms & Definitions
- Domain — The set of all input (x) values for which a function is defined and produces real outputs.
- Interval Notation — A notation for describing sets of numbers between endpoints on the real number line.
- Even Root — A root with an even index (such as a square root) which cannot have negative radicands in real numbers.
- Denominator — The bottom part of a fraction; must not be zero in real-valued functions.
Action Items / Next Steps
- Practice finding domains for various types of functions.
- Prepare questions on domain for next class.
- Review interval notation for homework.