BackGraphs of Basic Functions and Even/Odd Functions
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Section 1.2: Graphs of Basic Functions
Introduction to Basic Functions
Understanding the graphs of basic functions is foundational in calculus. These functions serve as building blocks for more complex mathematical models and are essential for interpreting and solving calculus problems.
Power Functions: Functions of the form where is an integer.
Reciprocal Functions: Functions of the form where is a positive integer.
Absolute Value Function: The function .
Key Basic Functions and Their Properties
(n odd, ): The graph passes through the origin, is symmetric about the origin, and increases more steeply as increases.
(n even, ): The graph is symmetric about the y-axis, passes through the origin, and opens upwards.
: The graph has vertical and horizontal asymptotes at and , respectively. It is symmetric about the origin.
: The graph has vertical and horizontal asymptotes at and , respectively. It is symmetric about the y-axis and always positive.
: The graph is V-shaped, symmetric about the y-axis, and passes through the origin.
Examples of Basic Functions
Example 1: is an odd power function. Its graph passes through the origin and is symmetric about the origin.
Example 2: is an even power function. Its graph passes through the origin, is symmetric about the y-axis, and opens upwards.
Example 3: is undefined at and has two branches in quadrants I and III.
Example 4: is undefined at and has two branches in quadrants I and II.
Example 5: is defined for all real and forms a V-shape.
Matching Functions to Graphs
When matching functions to their graphs, consider:
Symmetry (about the y-axis or origin)
Asymptotic behavior (vertical/horizontal asymptotes)
Domain and range
Even and Odd Functions
Definitions and Properties
Even Function: A function is even if for all in the domain. Property: The graph is symmetric about the y-axis.
Odd Function: A function is odd if for all in the domain. Property: The graph is symmetric about the origin.
Examples and Applications
Example 1: is an odd function because .
Example 2: is even but not odd because .
Example 3: is neither even nor odd because and .
Table: Comparison of Even and Odd Functions
Property | Even Function | Odd Function |
|---|---|---|
Definition | ||
Symmetry | y-axis | Origin |
Example | , | , |
Practice Problems
Show that is odd but not even.
Show that is even but not odd.
Determine all possible functions that are both even and odd. Hint: The only function is the zero function .
Additional info: Recognizing the symmetry of functions is crucial for graphing and for understanding their behavior under transformations, which is foundational for calculus topics such as integration and differentiation.
