Overview
This lecture reviews the definition, properties, and types of functions, discusses domains and ranges, and covers graphing basics, operations, composition, and function analysis, with practical examples.
Functions: Definition and Representation
- A function assigns each input (x, independent variable) to exactly one output (y, dependent variable) via a specified rule.
- The set of all possible inputs is the domain; the set of all possible outputs is the range.
- Function notation: y = f(x) or y = rule(x).
Basic Types of Functions and Graphs
- Linear function: Highest power of x is 1; graph is a straight line (e.g., f(x) = x).
- Quadratic function: Highest power is 2; graph is a parabola (e.g., f(x) = x²).
- Square root function: Graph starts at (0,0) and curves upward (e.g., f(x) = √x).
- Cubic function: Highest power is 3; graph is S-shaped (e.g., f(x) = x³).
- Exponential function: Rule involves a constant base to the x power (e.g., f(x) = 2Ë£).
- Logarithmic function: Inverse of exponential; has a vertical asymptote (e.g., f(x) = logâ‚‚x).
- Reciprocal function: Graph has two branches and asymptotes (e.g., f(x) = 1/x).
- Memorizing these shapes and their domains/ranges is key.
Domain and Range
- To determine domain: identify input restrictions (e.g., no negative under square root, no zero in denominator).
- To find range: graph the function and observe the lowest/highest y-values.
Evaluating and Composing Functions
- To evaluate, substitute the given input into the function rule (e.g., f(-2)).
- Function composition: Apply one function's output as input to another, e.g., g(f(x)).
Finding Intercepts and Zeros
- X-intercept: Set y = 0, solve for x.
- Y-intercept: Set x = 0, solve for y.
- Zeros: Inputs where f(x) = 0; coincide with x-intercepts.
Vertical Line Test & Function Properties
- A graph is a function if any vertical line intersects it at most once.
- Relations failing this test (e.g., circles) are not functions.
Combining Functions
- Sum, difference, product, and quotient of two functions are found by performing the operation on their rules.
Absolute Value and Piecewise Functions
- Absolute value graphs create a "V" shape; defined differently for positive and negative x.
- Piecewise functions are defined with different rules for different domains.
Increasing, Decreasing, and Constant Intervals
- A function is increasing where y-values rise as x increases, decreasing where y-values fall, and constant where y stays the same.
Key Terms & Definitions
- Domain — all possible input values for a function.
- Range — all possible output values from a function.
- Intercept — where a function crosses the x- or y-axis.
- Asymptote — a line the graph approaches but never touches.
- Vertical Line Test — a method to determine if a graph represents a function.
Action Items / Next Steps
- Complete assigned homework by showing all work on paper.
- Take photos of your completed work, merge into a PDF, and upload to Canvas.
- Review and memorize basic function graphs and domains/ranges for each.