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Study Guide - Smart Notes
Tailored notes based on your materials, expanded with key definitions, examples, and context.
Functions and Their Features
Definition of a Function
A function is a relation in which each input (independent variable) is assigned exactly one output (dependent variable). Functions can be represented by equations, tables, or graphs.
Vertical Line Test: A graph represents a function if and only if no vertical line intersects the graph at more than one point.
Notation: Functions are often written as , where is the input variable.
Example:
Given , this is a function because for each , there is exactly one .
Given , this is not a function of because some $x$-values correspond to two -values.
Features of a Graph
Key Vocabulary
Domain: The set of all possible input values () for the function .
Range: The set of all possible output values () for the function .
Independent Variable: The input variable, usually .
Dependent Variable: The output variable, usually .
x-intercept: The point(s) where the graph crosses the -axis (set ).
y-intercept: The point where the graph crosses the -axis (set ).
Intervals of Increase and Decrease
Increasing Interval: Where the function values rise as increases.
Decreasing Interval: Where the function values fall as increases.
Intervals of Positivity and Negativity
Positive Interval: Where (graph is above the -axis).
Negative Interval: Where (graph is below the -axis).
Analyzing Graphs and Tables
Example 1: Graph Analysis
Given a graph, identify the following features:
Domain:
Range:
x-intercept(s): and
y-intercept:
Interval where is increasing:
Interval where is decreasing:
Interval where is positive:
Interval where is negative: and
Example 2: Table Analysis
x | -2 | -1 | 0 | 1 | 2 |
|---|---|---|---|---|---|
f(x) | 4 | 1 | 0 | 1 | 4 |
Domain:
Range:
Interval where is decreasing:
Interval where is increasing:
Interval where is positive:
Interval where is negative: none
Example 3: Graph with Multiple Features
Domain:
Range:
x-intercept(s): ,
y-intercept:
Interval where is increasing: and
Interval where is decreasing:
Interval where is positive:
Interval where is negative: and
Piecewise Functions
Definition
A piecewise function is defined by different expressions for different intervals of the domain.
Example:
How to Analyze Piecewise Functions
Create a table of values for each interval.
Graph each piece on its respective interval.
Identify domain, range, intercepts, and intervals of increase/decrease/positivity/negativity.
Example:
Domain:
Range:
x-intercept(s):
y-intercept:
Interval where is positive:
Interval where is negative:
Interval where is increasing:
Interval where is decreasing:
Summary Table: Features of Functions
Feature | Definition | How to Find |
|---|---|---|
Domain | All possible -values | Look for input restrictions (e.g., denominators, square roots, graph endpoints) |
Range | All possible -values | Check graph or output values |
x-intercept(s) | Where | Solve |
y-intercept | Where | Compute |
Increasing Interval | rises as increases | Look for upward slope on graph |
Decreasing Interval | falls as increases | Look for downward slope on graph |
Positive Interval | Graph is above -axis | |
Negative Interval | Graph is below -axis |
Practice Problem Example
Given , graph the function and identify:
Domain:
Range:
x-intercept(s): and
y-intercept:
Interval where is increasing: and
Interval where is decreasing:
Interval where is positive:
Interval where is negative:
Additional info: These notes cover the identification of function features from equations, graphs, and tables, including domain, range, intercepts, and intervals of increase/decrease and positivity/negativity, with a focus on piecewise functions and graphical analysis.
