Overview
The lecture covers operations involving functions, focusing on addition, subtraction, multiplication, division, and composition, as well as factoring and simplifying polynomial expressions for function operations.
Operations on Functions
- Operations include addition, subtraction, multiplication, division, and composition of functions.
- When adding or subtracting functions, combine like terms by aligning similar powers of variables.
- For subtraction, change the sign of the second function before adding.
- Multiplying functions involves distributing each term in the first function to all terms in the second, adding exponents for like bases.
- Division and simplification require factoring both numerator and denominator and canceling common terms.
Factoring and Simplification
- Factor expressions to their simplest forms before performing operations or simplifying fractions.
- Recognize difference of squares: both terms must be perfect squares and the operation is subtraction.
- Factor trinomials by finding two numbers whose product is the constant term and whose sum is the middle coefficient.
- Use guess, check, and revise when factoring trinomials with leading coefficients other than 1.
- For common factors, factor out the greatest common divisor from all terms.
Polynomial Examples
- Always arrange polynomials in descending order of the variable’s exponent.
- When dividing, only cancel factors, not terms, from numerators and denominators after factoring.
- Combine like terms after addition, subtraction, or multiplication for a simplified result.
Composition of Functions
- Composition (f∘g)(x) means substitute g(x) into f(x) wherever x appears.
- Reverse order (g∘f)(x) substitutes f(x) into g(x).
- Nested compositions can involve more than two functions and require working from the innermost to the outermost.
Key Terms & Definitions
- Function — A relation where each input has one output.
- Composition of Functions — Applying one function to the results of another, written (f∘g)(x).
- Difference of Squares — An expression of the form a² - b² = (a + b)(a - b).
- Factoring — Rewriting an expression as a product of its factors.
- Like Terms — Terms with the same variable and exponent.
- Trinomial — A polynomial with three terms.
Action Items / Next Steps
- Practice addition, subtraction, and multiplication with given function examples.
- Factor and simplify given polynomials and function expressions.
- Complete assigned problems on function composition.
- Review factoring techniques for trinomials and difference of squares.