An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (2)
The function is a piecewise function with two parts: a quadratic 2 1 x 2 for x < 1 and a linear function 4 x − 1 for x ≥ 1 .
The range of the quadratic part is [ 0 , 2 1 ) .
The range of the linear part is [ 3 , ∞ ) .
The range of the entire piecewise function is the union of the two ranges: [ 0 , 2 1 ) ∪ [ 3 , ∞ ) . The final answer is [ 0 , 2 1 ) ∪ [ 3 , ∞ ) .
Explanation
Understanding the Piecewise Function We are given a piecewise function and asked to graph it and determine its range. The function is defined as: f ( x ) = { 2 1 x 2 4 x − 1 if if x < 1 x ≥ 1 We need to graph each piece of the function on the given domain and then find the range of the entire function.
Analyzing the First Piece First, let's analyze the first piece of the function, f ( x ) = 2 1 x 2 for x < 1 . This is a quadratic function, specifically a parabola. We know that at x = 0 , f ( 0 ) = 2 1 ( 0 ) 2 = 0 . As x approaches 1 from the left, f ( x ) approaches 2 1 ( 1 ) 2 = 2 1 . Since x < 1 , the value 2 1 is not included in the range of this piece. So, the range of this piece is [ 0 , 2 1 ) .
Analyzing the Second Piece Next, let's analyze the second piece of the function, f ( x ) = 4 x − 1 for x ≥ 1 . This is a linear function. At x = 1 , f ( 1 ) = 4 ( 1 ) − 1 = 3 . As x increases from 1, f ( x ) also increases. Since there is no upper bound on x , there is no upper bound on f ( x ) . So, the range of this piece is [ 3 , ∞ ) .
Combining the Ranges Now, we combine the ranges of the two pieces to find the range of the entire function. The range of the first piece is [ 0 , 2 1 ) and the range of the second piece is [ 3 , ∞ ) . Therefore, the range of the entire function is the union of these two intervals: [ 0 , 2 1 ) ∪ [ 3 , ∞ ) .
Final Answer The range of the piecewise function is [ 0 , 2 1 ) ∪ [ 3 , ∞ ) .
Examples
Piecewise functions are used in real life to model situations where different rules or conditions apply over different intervals. For example, cell phone billing plans often have different rates for data usage depending on whether you are below or above a certain data limit. Similarly, income tax brackets are a piecewise function where different tax rates apply to different income ranges. Understanding piecewise functions helps in analyzing and predicting outcomes in these scenarios.
In a device delivering a current of 15.0 A for 30 seconds, around 2.81 × 1 0 21 electrons flow through it. This is calculated by first finding the total charge using the formula Q = I ⋅ t , then dividing that charge by the charge of a single electron ( 1.602 × 1 0 − 19 C ). Thus, approximately 2.81 × 1 0 21 electrons pass through the device during that time period.
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