Relations and Functions Worksheet: A Comprehensive Guide with Answers
This worksheet provides a thorough exploration of relations and functions, crucial concepts in algebra and beyond. It's designed to help you solidify your understanding through a range of exercises, with detailed answers provided at the end. Whether you're a student looking to improve your understanding or a teacher seeking supplementary materials, this worksheet offers a valuable resource.
What are Relations and Functions?
Before diving into the exercises, let's briefly review the definitions:
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Relation: A relation is a set of ordered pairs (x, y). Essentially, it shows a connection or correspondence between elements of two sets. Not all relations are functions.
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Function: A function is a special type of relation where each input (x-value) corresponds to exactly one output (y-value). This means no two ordered pairs can have the same x-value but different y-values. We often represent functions using function notation, such as f(x) = ...
Worksheet Exercises:
Section 1: Identifying Relations and Functions
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Determine if each relation is a function:
a) {(1, 2), (2, 4), (3, 6), (4, 8)} b) {(1, 2), (2, 4), (1, 6), (4, 8)} c) {(1, 2), (2, 2), (3, 2), (4, 2)} d) {(x, y) | y = x²} (Consider the graph of this relation)
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Given the following mappings, state whether they represent functions:
a) A mapping from the set of students in a class to their ages. b) A mapping from the set of positive integers to their square roots. c) A mapping from the set of states in the US to their capitals.
Section 2: Function Notation and Evaluation
Given the function f(x) = 2x + 3, find:
- f(2)
- f(-1)
- f(0)
- f(a)
- f(x + 1)
Section 3: Domain and Range
Determine the domain and range of the following functions:
- f(x) = x²
- f(x) = √x
- f(x) = 1/(x - 2)
- f(x) = |x|
Section 4: Graphing Functions
Sketch the graphs of the following functions:
- f(x) = x
- f(x) = x³
- f(x) = |x|
- f(x) = 1/x (Hyperbola)
Answers:
(Note: Detailed explanations and graphical representations would normally be included here for each question. Due to the limitations of this markdown format, providing sketches is not feasible. However, the answers are provided below to guide you in solving the problems.)
Section 1:
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a) Function b) Not a function c) Function d) Function
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a) Function b) Function c) Function
Section 2:
- f(2) = 7
- f(-1) = 1
- f(0) = 3
- f(a) = 2a + 3
- f(x + 1) = 2x + 5
Section 3:
- Domain: All real numbers; Range: All non-negative real numbers.
- Domain: All non-negative real numbers; Range: All non-negative real numbers.
- Domain: All real numbers except x = 2; Range: All real numbers except y = 0.
- Domain: All real numbers; Range: All non-negative real numbers.
Section 4:
(Graphs should be sketched – refer to a graphing calculator or online graphing tool to visualize these functions).
This worksheet serves as a strong foundation for understanding relations and functions. Remember to practice regularly and consult your textbook or teacher if you require further assistance. Mastering these concepts is key to success in higher-level mathematics.
