뒤로Graphs and Coordinates: Functions, Domains, and Intercepts
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Graphs and Coordinates
Functions and Relations
Understanding the distinction between relations and functions is fundamental in precalculus. A relation is any set of ordered pairs, while a function is a special type of relation where each input (domain value) corresponds to exactly one output (range value).
Definition: A function from set A to set B assigns each element of A to exactly one element of B.
Domain: The set of all possible input values (x-values).
Range: The set of all possible output values (y-values).
Example: For the relation {(1, 7), (2, 8), (3, 7)}, the domain is {1, 2, 3} and the range is {7, 8}.
Evaluating Functions
To evaluate a function, substitute the given value of the independent variable into the function's formula.
Example: If , then .
Application: Function evaluation is used to determine specific outputs for given inputs, which is essential for graphing and analysis.
Difference Quotient
The difference quotient is a formula that gives the average rate of change of a function over an interval. It is foundational for understanding derivatives in calculus.
Formula:
Example: If , then:
So,
Evaluating and Simplifying Functions
When given a function and a specific input, substitute and simplify as required.
Example: If , then
Equations Defining Functions
To determine if an equation defines y as a function of x, solve for y and check if each x-value yields only one y-value.
Example: For , if [y] denotes the greatest integer function, further analysis is needed. If it is a linear equation, it can be solved for y, indicating y is a function of x.
Key Point: If for every x there is only one y, then y is a function of x.
Intercepts of Graphs
Intercepts are points where a graph crosses the axes. The x-intercept is where the graph crosses the x-axis (y = 0), and the y-intercept is where it crosses the y-axis (x = 0).
Finding x-intercepts: Set y = 0 and solve for x.
Finding y-intercepts: Set x = 0 and solve for y.
Example: If a line crosses the x-axis at (2, 0) and the y-axis at (0, -6), these are the intercepts.
Summary Table: Key Concepts
Concept | Definition | Example |
|---|---|---|
Function | Each input has exactly one output | {(1, 2), (2, 3)} |
Domain | Set of all possible x-values | {1, 2, 3} |
Range | Set of all possible y-values | {2, 3} |
Difference Quotient | Average rate of change | |
Intercepts | Where graph crosses axes | (2, 0), (0, -6) |
