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Representing Functions

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Understand that a function from one set (the input, called the domain) to another set (the output, called the range) assigns to each element of the domain exactly one element of the range, i.e. each input value maps to exactly one output value. If f is a function, x is the input (an element of the domain), and f(x) is the output (an element of the range). Graphically, the graph is y = f(x).

Modeling with Functions

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Decide which type of function is most appropriate by observing graphed data, charted data, or by analysis of context to generate a viable (rough) function of best fit. Use this function to solve problems in context. Emphasize linear, quadratic and exponential models.

Linear Functions and Their Inverses

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Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

Operations with Linear Function

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Write a function that describes a relationship between two quantities. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.

Writing Linear Functions

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Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.  Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

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Write a function that describes a relationship between two quantities.

Using tables, graphs, and verbal descriptions, interpret the key characteristics of a function which models the relationship between two quantities. Sketch a graph showing key features including: intercepts; interval where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

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Write a function that describes a relationship between two quantities.  Create linear, quadratic, and exponential equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

Discrete Linear Functions

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Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.  Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

Discrete Exponential Functions

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Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.  Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.

Piecewise Functions

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Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.