Overview
This lecture introduces the concepts of functions, equations in two variables, function notation, domains and ranges, and their applications in business contexts like profit-loss and price-demand analysis.
Equations in Two Variables and Graphing
- Linear equations (Ax + By = C) graph as lines; each solution is a point on the line.
- More complex equations like y = 9x² require point-by-point plotting using tables of values.
Functions and Correspondences
- A function is a correspondence where each domain element matches exactly one range element.
- The domain is the set of all possible inputs; the range is the set of all possible outputs.
- Not all equations give functions—if any input gives more than one output, it's not a function.
Determining Functions and the Vertical Line Test
- The vertical line test: a graph represents a function if no vertical line crosses it more than once.
- If an equation in two variables gives one output for each input, it specifies a function.
Domains and Ranges
- The domain of a function is all real input values producing real outputs.
- To find a function's domain, exclude values that make the output undefined (e.g., division by zero, negative roots).
Function Notation and Evaluation
- f(x) replaces y to show the output for a specific input, x.
- To evaluate, substitute the value into the function: e.g., f(2), g(1), h(8).
- Example: For f(x) = x² – 2x + 7, find f(a), f(a + h), and the difference quotient.
The Difference Quotient
- The difference quotient is [f(x + h) – f(x)] / h and is foundational in calculus for finding rates of change.
Business Applications: Profit-Loss and Price-Demand
- Cost (C), revenue (R), and profit (P) functions model business scenarios.
- C(x) = fixed cost + variable cost (often C(x) = a + bx).
- Revenue: R(x) = x * p(x), where p(x) is the price-demand function (e.g., p(x) = m – n x).
- Profit: P(x) = R(x) – C(x).
- Break-even occurs when R = C; profit is positive when R > C.
Key Terms & Definitions
- Function — A relation where each input (domain) has exactly one output (range).
- Domain — Set of all possible input values for a function.
- Range — Set of all possible outputs for a function.
- Vertical Line Test — A method to determine if a graph represents a function.
- Difference Quotient — [f(x + h) – f(x)] / h; measures average rate of change.
- Price-Demand Function — Shows relationship between price and quantity sold.
- Profit Function — P(x) = revenue – cost.
Action Items / Next Steps
- Complete matched problems from lecture, especially on identifying functions and finding domains.
- Practice graphing price-demand, revenue, and profit functions.
- Review function evaluation and difference quotient calculations for provided examples.