BackFunctions and Linear Functions: Foundations and Evaluation
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Functions and Linear Functions
Introduction to Functions
Functions are a fundamental concept in algebra, describing relationships between sets of values. Understanding functions involves identifying domains and ranges, determining whether a relation is a function, and evaluating functions using equations or tables.
Definition of a Relation
Relation: Any set of ordered pairs (x, y).
Domain: The set of all first components (x-values) in the ordered pairs.
Range: The set of all second components (y-values) in the ordered pairs.
Example: Given the set {(golf, 250), (lawn mowing, 325), (water skiing, 430), (hiking, 430), (bicycling, 720)}, the domain is {golf, lawn mowing, water skiing, hiking, bicycling} and the range is {250, 325, 430, 720}.
Definition of a Function
Function: A correspondence from a domain to a range such that each element in the domain corresponds to exactly one element in the range.
Key Point: If any element in the domain is paired with more than one element in the range, the relation is not a function.
Determining Whether a Relation Is a Function
Check if any domain value is associated with more than one range value.
If so, the relation is not a function; otherwise, it is a function.
Example 2a: {(1, 2), (3, 4), (5, 6), (5, 8)} Here, 5 corresponds to both 6 and 8, so this is not a function.
Example 2b: {(1, 2), (3, 4), (6, 5), (8, 5)} Each domain value corresponds to exactly one range value, so this is a function.
Functions as Equations and Function Notation
Functions are often represented by equations, where the output variable (commonly y) is a function of the input variable (commonly x). Function notation uses letters such as f, g, or h to name functions.
Function Notation: f(x) represents the value of the function f at x.
For each value of x, there is exactly one value of f(x).
Example: For f(x) = 4x + 5, f(6) is found by substituting 6 for x:
Example: For g(x) = 3x^2 - 10, g(-5) is:
Functions Defined by Tables
Functions can also be represented by tables, where each input value (domain) is paired with exactly one output value (range).
If every input value corresponds to only one output value, the table defines a function.
Example Table:
Input (x) | Output (g(x)) |
|---|---|
0 | 3 |
1 | 0 |
2 | 1 |
3 | 2 |
4 | 3 |
Domain: {0, 1, 2, 3, 4} Range: {0, 1, 2, 3}
Evaluating Function Values from a Table:
g(1) = 0
g(3) = 2
Find x such that g(x) = 3: x = 0 or x = 4
