Overview
This lecture covers the definition, properties, and types of functions, how to evaluate and combine them, analyze their domains and ranges, determine intercepts and zeros, perform function composition, and identify intervals of increase, decrease, or constancy.
Basics of Functions
- A function applies a rule to an input (independent variable, x) to produce an output (dependent variable, y or f(x)).
- The domain is the set of all possible inputs; the range is the set of all possible outputs.
- If a function's highest power of x is one, it's a linear function and its graph is a straight line.
Common Types of Functions and Graphs
- Linear: f(x) = x — graph is a straight line through the origin.
- Quadratic: f(x) = x² — graph is a parabola opening upwards; domain: (-∞, ∞); range: [0, ∞).
- Square Root: f(x) = √x — only defined for x ≥ 0; graph increases slowly; domain/range: [0, ∞).
- Cubic: f(x) = x³ — graph is an S-shaped curve; domain/range: (-∞, ∞).
- Exponential: f(x) = 2ˣ — graph grows rapidly, horizontal asymptote at y=0; domain: (-∞, ∞); range: (0, ∞).
- Logarithmic: f(x) = log₂x — graph passes through (1,0), vertical asymptote at x=0; domain: (0, ∞); range: (-∞, ∞).
- Reciprocal: f(x) = 1/x — two branches, asymptotes at x=0 and y=0; domain/range: (-∞,0)∪(0,∞).
Evaluating and Composing Functions
- To evaluate, substitute the given value for x in the function.
- Composition: (g∘f)(x) = g(f(x)), meaning output of f is used as input for g.
Domain and Range Analysis
- Analyze function operations to determine restrictions (e.g., square root requires non-negative input, division by zero is undefined).
- Use graph transformations to determine range after shifts.
Intercepts and Zeros
- X-intercept: set y=0 and solve for x.
- Y-intercept: set x=0 and solve for y.
- Zeros of a function: values of x where f(x)=0; same as x-intercepts.
Vertical Line Test
- A graph represents a function if no vertical line intersects it more than once.
Combining Functions
- Functions can be added, subtracted, multiplied, or divided, combining their expressions accordingly.
Piecewise and Absolute Value Functions
- Absolute value: positive inputs unchanged; negative inputs become positive; forms a "V" shape.
- Piecewise: different expressions for different input intervals.
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-values do not change.
Key Terms & Definitions
- Function — a rule assigning each input exactly one output.
- Domain — all possible inputs (x-values).
- Range — all possible outputs (y-values).
- Linear Function — function of the form f(x)=mx+b.
- Parabola — graph of a quadratic function.
- Asymptote — a line a graph approaches but never touches.
- Intercept — point where the graph crosses an axis.
- Zero of a Function — input where the output is zero.
- Piecewise Function — defined by different rules on different intervals.
- Vertical Line Test — test to determine if a graph is a function.
Action Items / Next Steps
- Complete and show all work for assigned paper-pencil homework.
- Submit photos of work as a PDF via Canvas as instructed.