BackFunctions: Definitions, Examples, and Determining Functionality
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Section 2.1: Functions
Introduction to Relations and Functions
Understanding the concept of functions is foundational in precalculus. This section introduces the definitions of relations and functions, explains the concepts of domain and range, and provides examples to determine whether a relation is a function.
Definitions
Relation: A relation is a correspondence between two sets, typically denoted as X and Y.
Function: A function from X to Y is a relation that associates with each element of X exactly one element of Y.
Domain: The set of all values on which the function is defined (inputs).
Range: The set of all outputs (images) of the function.
Visual Representation
A function can be visualized as a mapping from each element in set X (domain) to exactly one element in set Y (range). If any element in X is mapped to more than one element in Y, the relation is not a function.
Examples and Applications
Example 1: Table of Values
Person | Age |
|---|---|
Alex | 9 |
Ellie | 6 |
Nat | 9 |
Grandma | 70 |
Domain: {Alex, Ellie, Nat, Grandma}
Range: {6, 9, 70}
Conclusion: This is a function because each person is associated with exactly one age.
Example 2: Non-Function Table
Animal | Appendage |
|---|---|
Fish | Fins |
Dog | Legs |
Bird | Legs, Wings |
For the element "bird," the relation associates two different y values (legs, wings). Therefore, this is not a function.
Additional info: A function cannot assign more than one output to a single input.
Example 3: Ordered Pairs
Determine if the relation is a function.
a) { (2, 3), (4, 1), (3, 2), (2, 1) } Not a function. The value 2 is associated with two different y values (3 and 1).
b) { (-2, 3), (4, 1), (3, -2), (2, 1) } This is a function. Each input has only one output. Domain: { -2, 2, 3, 4 } Range: { -2, 1, 3 }
Functions Defined by Equations
To determine if an equation defines y as a function of x, check if for every input x there is only one output y.
a) Yes, this is a function. For any input x, there is only one output y.
b) Solve for y:
Yes, this is a function. For each x, there is only one y value.
c) Solve for y:
Not a function! For some values of x, there are two possible y values.
Key Points and Properties
A function assigns exactly one output to each input.
It is acceptable for different inputs to have the same output.
If any input is associated with more than one output, the relation is not a function.
Additional Info
When analyzing tables, ordered pairs, or equations, always check if each input corresponds to only one output to determine if the relation is a function.
Functions are foundational for understanding more advanced topics in mathematics, including calculus and analytic geometry.
