Understanding Functions and Relations
What is a Relation?
- A relation is any set of ordered pairs, which can be graphed.
- Domain: The first component of the ordered pair (x-values).
- Range: The second component of the ordered pair (y-values).
- To list domain and range:
- Domain: {0, 10, 20, 30, 40}
- Range: {6.7, 9.1, 10.7, 13.2, 21.2}
What is a Function?
- A function is a correspondence from a first set to a second set where each element in the domain corresponds to exactly one element in the range.
- Key Point: No x-values can be repeated.
- Example:
- Set 1: x-values {1, 3, 6, 8} - all different, so it is a function.
- Set 2: x-values {2, 2, 3} - 2 is repeated, so it is not a function.
Determining if an Equation is a Function
- Solve for y: If more than one y-value can be obtained, it is not a function.
- Example: Subtracting x² and taking the square root introduces ± (means not a function).
Function Notation
- Basic Concept: Rewrite y = x + 2 as f(x) = x + 2.
- f(x) is equivalent to y in equations.
- Finding f(x):
- Example: f(-5)
- Calculation: (Negative 5)² = 25, resulting calculation gives y = 42.
- Point on graph: (-5, 42)
Examples with Variables and Expressions
- Finding f(3a):
- Plug in 3a for x, compute to result in 9a² + 6a + 7.
- Finding f(4 + z):
- Plug in 4 + z for x, expand and simplify to z² + 6z + 15.
Graphing Functions
- Graph of a Function:
- Example: Graph y = 2x and y = 2x - 3.
- Determine y for various x-values, plot points, and draw lines.
Determining Domain and Range from Graphs
- Domain: The set of all x-values on the graph.
- Use brackets [ ] for closed/inclusive points, parenthesis ( ) for open/exclusive points.
- Range: The set of all y-values.
- Example with solid and open dots:
- Domain: [-2, 1]
- Range: [0, 3]
- Open dot at an endpoint: Use parenthesis.
Finding Intercepts
- X-intercepts: Points where the graph crosses the x-axis.
- Example: (-3, 0), (-1, 0), (2, 0)
- Y-intercepts: Points where the graph crosses the y-axis.
This concludes the section on understanding functions, relations, and how to interpret and graph them.