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Understanding Functions

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Explore the fundamental concepts of functions in mathematics, including their definition as unique associations between sets, and their essential characteristics like domain and range. Functions are depicted through various graphical forms, such as lines for linear functions and parabolas for quadratic functions. The text delves into function evaluation, graphical interpretation, and advanced functions like exponential and logarithmic types.

Fundamental Concepts of Functions in Mathematics

In mathematics, a function is a relation that uniquely associates each element of a given set, known as the domain, with exactly one element of another set, often called the codomain. A function can be thought of as a rule that assigns to each input exactly one output. This relationship is typically written as f(x), where 'x' is the input variable and 'f(x)' represents the output. The defining characteristic of a function is that no input in the domain corresponds to more than one output in the codomain.
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Essential Characteristics of Functions

Understanding functions requires familiarity with their essential characteristics. The 'domain' consists of all the allowable inputs, while the 'range' is the set of all possible outputs. The input, or independent variable, is what we manipulate to observe changes in the output, or dependent variable. For example, in a function that models the growth of a rabbit population over time, 'time' is the independent variable, and the 'number of rabbits' is the dependent variable. Graphical representations, such as function diagrams, are instrumental in visualizing the relationship between inputs and outputs.

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00

A function is often symbolized as ______, where 'x' signifies the ______ and '______' denotes the resulting output.

f(x)

input variable

f(x)

01

Define 'domain' in functions.

Domain: Set of all allowable inputs for a function.

02

Define 'range' in functions.

Range: Set of all possible outputs from a function.

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