Express the monthly cost $E$ as a piecewise defined function of $x$. Assume $E$ is measured in dollars.

$E(x)=\left\{\begin{array}{ll} / /, & \text { if } 0 \leq x \leq 300 \\ / /, & \text { if } 300<x \leq 600 \end{array}\right.$

(b) Graph the function $E$ for $0 \leq x \leq 600$.

The monthly cost E as a piecewise defined function of x is 300 \end{cases}"> E ( x ) = { 0.10 x , ​ if 0 ≤ x ≤ 300 0.20 x − 30 , ​ if x > 300 ​ . The graph consists of two line segments: one from ( 0 , 0 ) to ( 300 , 30 ) and another from ( 300 , 30 ) to ( 600 , 90 ) . 300 \end{cases}}"> E ( x ) = { 0.10 x , ​ if 0 ≤ x ≤ 300 0.20 x − 30 , ​ if x > 300 ​ ​
Explanation

Understanding the Problem We are given a problem about calculating the monthly electricity cost E based on the amount of electricity used, x kWh. The cost is determined by a piecewise function: a fixed cost for the first 300 kWh and an additional cost of $0.20 per kWh for usage exceeding 300 kWh. We need to express this as a piecewise function and then describe how to graph it.

Expressing the Cost For 0 ≤ x ≤ 300 , the cost is simply 0.10 x . For 300"> x > 300 , the cost is 0.10 ( 300 ) for the first 300 kWh, plus $0.20 for each kWh over 300. So the cost is 0.10 ( 300 ) + 0.20 ( x − 300 ) .

Piecewise Function Therefore, the piecewise function is


300 \end{cases}"> E ( x ) = { 0.10 x , ​ if 0 ≤ x ≤ 300 0.10 ( 300 ) + 0.20 ( x − 300 ) , ​ if x > 300 ​
Simplifying the second part of the piecewise function:
300 \end{cases}"> E ( x ) = { 0.10 x , ​ if 0 ≤ x ≤ 300 30 + 0.20 x − 60 , ​ if x > 300 ​
300 \end{cases}"> E ( x ) = { 0.10 x , ​ if 0 ≤ x ≤ 300 0.20 x − 30 , ​ if x > 300 ​

Graphing the Function To graph the function, we plot E ( x ) = 0.10 x for 0 ≤ x ≤ 300 , which is a line with slope 0.10 starting from the origin. At x = 300 , E ( 300 ) = 0.10 ( 300 ) = 30 .

For 300"> x > 300 , we plot E ( x ) = 0.20 x − 30 . This is a line with slope 0.20. At x = 300 , E ( 300 ) = 0.20 ( 300 ) − 30 = 60 − 30 = 30 . So the two parts of the function connect at ( 300 , 30 ) .
To graph for 0 ≤ x ≤ 600 , we need to evaluate E ( 600 ) = 0.20 ( 600 ) − 30 = 120 − 30 = 90 . So the line segment will go from ( 300 , 30 ) to ( 600 , 90 ) .
The graph will be a piecewise linear function.
Examples
Understanding piecewise functions is crucial in many real-world scenarios, such as calculating income taxes, where different income brackets are taxed at different rates. Similarly, shipping costs often follow a piecewise function, with different rates applying to different weight ranges. In manufacturing, production costs might have a fixed setup cost plus a variable cost per unit, creating a piecewise cost function. By mastering piecewise functions, you can model and analyze these situations effectively.

Answered by GinnyAnswer | 2025-07-08

The monthly cost E can be expressed as a piecewise function: E ( x ) = { 0.10 x , ​ if 0 ≤ x ≤ 300 0.20 x − 30 , ​ if 300 < x < 600 ​ . The graph consists of two line segments, one from ( 0 , 0 ) to ( 300 , 30 ) and another from ( 300 , 30 ) to ( 600 , 90 ) , indicating how the cost changes with electricity consumption.
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Answered by Anonymous | 2025-07-11