Chapter Outline
Algebra often feels like a puzzle where both the problem and the solution are numbers. In this chapter, we will explore how algebra can model complex real-world situations with just two variables. You’ll start by mastering the simplification of algebraic expressions, learning to identify components and apply the order of operations effectively. Moving deeper, you will write and solve linear equations in two variables, uncovering unknown quantities and establishing relationships grounded in practical scenarios.
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By the end of this chapter, you’ll not only manipulate equations with ease but also apply your understanding to decode complex scenarios, whether in academics or everyday life, using the powerful tool of algebra.
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3.1
Introduction to Functions
Learning Objectives
By the end of this section, you will be able to:
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define a function in terms of its input and output, and identify whether a relation is a function by checking if every element of the domain is paired with exactly one element of the range.
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determine the domain and range of a function from graphs, tables, and equations, understanding these as sets of possible input and output values.
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create and interpret input/output tables for functions, using them to explore how changes in input affect the output.
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analyze and interpret the graphs of functions, applying this understanding to real-world scenarios to determine viable solutions.
3.2
Function Notation
Learning Objectives
By the end of this section, you will be able to:
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explain what function notation is and how it replaces the dependent variable with f(x) , where f represents the function’s value at a given independent value x .
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write and interpret expressions using function notation to describe relationships between variables in mathematical and real-world contexts.
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use function notation to find outputs for specific inputs by substituting values into functions and solving.
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translate given relationships, such as equations or ordered pairs, into function notation, clearly identifying the input and output components.
3.3
Inverse Functions
Learning Objectives
By the end of this section, you will be able to:
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explain what an inverse function is and understand how it relates to the concept of reversing the effects of a function.
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find the inverse of a given function, ensuring that the function and its inverse undo each other.
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graph a function and its inverse, identifying symmetries and the line y = x as a reflection axis.
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apply inverse functions to solve real-world problems by reversing processes to find original values from given outcomes.
3.4
Evaluating Functions
Learning Objectives
By the end of this section, you will be able to:
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evaluate the output of a function for given input values using function notation.
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determine possible input values when given specific output values from a function.
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use graphs to evaluate functions and find output values corresponding to specific inputs.
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work with functions represented in tables, graphs, and symbolic forms to find and verify outputs and inputs.
3.5
Solving Absolute Value Functions
Learning Objectives
By the end of this section, you will be able to:
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explain the concept of absolute value as the distance from zero on a number line and understand how it is always positive.
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write absolute value equations from real-world scenarios to model situations like communication range or error margins.
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solve absolute value equations, converting them into equivalent linear equations to find both possible solutions.
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analyze the solutions of absolute value equations and compare their properties to those of linear equations.