Understanding Functions and Relations
Introduction to Functions
- Function Notation: Represents a function by using a symbol like
f(x). This notation is interchangeable with y in equations if the equation is a function.
- Writing Equations in Function Notation:
- A linear equation
y = 3x + 1 can be rewritten as f(x) = 3x + 1.
- A quadratic equation
y = 3(x - 7)^2 + 5 becomes f(x) = 3(x - 7)^2 + 5.
Purpose of a Function
- Function as a Machine: Conceptualize a function as a machine where each input (x-value) produces a single output (y-value).
- Example:
- For
f(x) = 3x + 1, if x=2, then f(2) = 7. If x=5, then f(5) = 16.
Function vs. y-Notation
- Both
y = 3x + 1 and f(x) = 3x + 1 result in the same outputs when evaluated, e.g., f(4) = 13.
- Benefits of Function Notation:
- Clearly indicates the expression represents a function.
- Provides immediate information about specific inputs and outputs.
Identifying a Function
- Criteria: A function has each x-value producing only one y-value.
- Example of Non-Function:
y^2 = x is not a function because, for a single x-value like 9, it provides two y-values (+3, -3).
- Relation: Any set of coordinates; broader than functions.
- Every function is a relation, but not every relation is a function.
The Vertical Line Test
- Purpose: A graphical method to determine if a curve is a function.
- How it Works:
- Draw vertical lines on the graph.
- If any vertical line crosses more than one point, it is not a function.
- Example: A circle fails the vertical line test, whereas a parabola and a linear graph pass.
Horizontal Lines and Functions
- A function allows multiple x-values to map to a single y-value, hence horizontal lines crossing multiple points does not disqualify a function.
Conclusion
- Functions are specific types of relations where each x-value maps to only one y-value.
- The vertical line test is a useful tool for visually verifying whether a graph represents a function.
These notes summarize key concepts about functions, relations, and the vertical line test. Understanding these basics is crucial for advanced mathematical exploration.