Relations and Functions Worksheet: A Comprehensive Guide with Answers
This worksheet provides a comprehensive exploration of relations and functions, crucial concepts in algebra and beyond. We'll delve into identifying relations, determining whether a relation is a function, and working with function notation. Each section includes practice problems with detailed solutions at the end. This worksheet is designed to be printed as a PDF for convenient offline use.
Section 1: Understanding Relations
A relation is simply a set of ordered pairs (x, y). These pairs represent a connection or association between elements from two sets, often denoted as the domain and range. The domain is the set of all possible x-values, and the range is the set of all possible y-values.
Example: The relation {(1, 2), (3, 4), (5, 6)} has a domain of {1, 3, 5} and a range of {2, 4, 6}.
Practice Problems:
- Identify the domain and range of the relation {(0, 1), (1, 1), (2, 4), (3, 9)}.
- Represent the relation where each x-value is the square of its corresponding y-value. Use at least four ordered pairs.
Section 2: Identifying Functions
A function is a special type of relation where each x-value (input) is associated with exactly one y-value (output). In other words, for every x, there's only one corresponding y. We can use the vertical line test graphically: if a vertical line intersects the graph at more than one point, it's not a function.
Practice Problems:
- Determine whether the following relations are functions: a) {(1, 2), (2, 3), (3, 4), (4, 5)} b) {(1, 2), (2, 3), (1, 4), (3, 5)} c) {(x, y) | y = x²} (Hint: Consider the graph of y = x²)
- Explain why the equation x = y² does not represent a function.
Section 3: Function Notation
Function notation uses the symbol f(x) (read as "f of x") to represent the output of a function f for a given input x. This notation is concise and powerful.
Example: If f(x) = 2x + 1, then f(3) = 2(3) + 1 = 7.
Practice Problems:
- Given f(x) = x² - 3x + 2, find: a) f(0) b) f(2) c) f(-1)
- If g(x) = 4x - 5, find the value of x such that g(x) = 7.
Section 4: Domain and Range of Functions
Determining the domain and range is crucial for understanding a function's behavior. The domain often includes all real numbers except those that lead to undefined operations (like division by zero or even roots of negative numbers). The range is determined by the possible outputs.
Practice Problems:
- Find the domain and range of the function f(x) = √(x - 4). (Hint: Consider what values of x lead to a real number output.)
- Find the domain of the function g(x) = 1/(x + 2).
Answer Key:
- Domain: 0, 1, 2, 3}; Range
- Answers may vary, examples include {(1, 1), (4, 2), (9, 3), (16, 4)}
- a) Function; b) Not a function; c) Function
- For some x values (e.g., x = 4), there are two corresponding y values (y = 2 and y = -2). Fails the vertical line test.
- a) 2; b) 0; c) 6
- x = 3
- Domain: x ≥ 4; Range: y ≥ 0
- Domain: All real numbers except x = -2
This worksheet offers a solid foundation in understanding relations and functions. Remember to practice consistently to master these vital concepts. Further exploration can include piecewise functions, inverse functions, and more advanced function operations.
