An electricity company charges its customers a base rate of ...

An electricity company charges its customers a base rate of $$\$ 10$$ a month, plus 6 cents per kilowatt-hour ( kWh ) for the first 1200 kWh and 7 cents per kWh for all usage over 1200 kWh . Express the monthly cost $E$ as a function of the amount $x$ of electricity used. Then graph the function $E$ for $0 \leqslant x \leqslant 2000$.

Key Concepts

Piecewise Functions

A piecewise function is a function defined by multiple sub-functions, each of which applies to a specific interval in the domain. This concept is essential when a situation involves different rules or formulas for different ranges of inputs, such as tiered pricing structures.

Linear Functions

Linear functions are characterized by a constant rate of change and are typically expressed in the form y = mx + b, where m is the slope and b is the y-intercept. In this context, the cost calculation for each tier of electricity usage is represented by linear functions.

Domain and Range

The domain of a function refers to the set of all possible input values, while the range is the set of output values produced by the function. Specifying the domain, such as electricity usage from 0 to 2000 kWh, ensures that the model accurately reflects the meaningful values in the real-world scenario.

Graphing Functions

Graphing a function involves plotting the input-output relationship on a coordinate plane. For piecewise functions, this means plotting each segment over its respective interval, which often results in a graph with distinct linear segments that change at the points where the function's rule changes.

Function Modeling

Function modeling is the process of creating a mathematical representation of a real-world process. In this case, it involves translating a billing scheme—consisting of a base rate plus varying per-unit charges—into a mathematical function that accurately reflects the costs for different levels of electricity usage.

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