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1.0: Introduction to Functions
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This is an introduction to the first chapter of the course. In this chapter, we review of the basic concepts behind functions and gather all the tools we will need for understanding and working with future material.
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1.1: Functions and Function Notation
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This section introduces the concept of functions, focusing on their definition and notation. It explains how to identify a function, understand domain and range, and use function notation such as \( f(x) \). The section also includes examples of evaluating functions, interpreting their meanings in various contexts, and distinguishing functions from general relations.
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1.2: Domain and Range
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This section explores the concepts of domain and range for functions. It defines the domain as the set of all possible input values and the range as the set of all possible output values. The section provides techniques for determining domain and range from equations, graphs, and real-world contexts. Examples illustrate how to identify restrictions on the domain, such as division by zero or square roots of negative numbers.
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1.3: The Arithmetic and Composition of Functions
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Combining two relationships into one function, we have performed function composition, which is the focus of this section. Function composition is only one way to combine existing functions. Another way is to carry out the usual algebraic operations on functions, such as addition, subtraction, multiplication and division. We do this by performing the operations with the function outputs, defining the result as the output of our new function.
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1.4: Behavior of Graphs of Functions
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This section analyzes the behavior of function graphs, including identifying intervals of increase or decrease, local maxima and minima, and symmetry. It also explores how to calculate and interpret the average rate of change and examine extrema using graphing techniques. Examples demonstrate analyzing and sketching functions based on these behaviors.
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1.5: Transformation of Functions
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This section explores transformations of functions, including vertical and horizontal shifts, reflections, stretches, and compressions. It explains how changes to the function's equation affect its graph and provides examples to illustrate each type of transformation. These concepts help analyze and modify function behavior visually and algebraically.
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1.6: Absolute Value Functions
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This section examines absolute value functions, focusing on their definition, properties, and graphing. It explains how to solve absolute value equations and inequalities and interpret their solutions. The section includes examples that illustrate shifts, reflections, and stretches of absolute value graphs, helping readers understand their behavior and applications.
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1.7: Inverse Functions
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This section explores inverse functions, explaining how to determine if a function has an inverse and how to find it. It covers verifying inverses by composition, graphing inverses as reflections over the line \(y = x\), and restricting domains to ensure functions are invertible. Examples illustrate these concepts for various types of functions.
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1.8: Chapter Review
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