An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (2)
Check if g ( x ) = 4 ( x − 1 ) : This is false because g ( 0 ) = 0 in the table, but 4 ( 0 − 1 ) = 4 1 .
Check if g ( x ) = 4 x − 1 : This is true because the values match the table.
Determine the graph translation: Since g ( x ) = f ( x ) − 1 , the graph is translated down 1 unit.
Compare domain and range: The domain of both functions is all real numbers, but the range of f ( x ) is ( 0 , ∞ ) and the range of g ( x ) is ( − 1 , ∞ ) .
The two true statements are g ( x ) = 4 x − 1 and the graph is translated down 1 unit.
Explanation
Analyzing the Problem We are given a table of values for two functions, f ( x ) and g ( x ) , and we know that f ( x ) = 4 x . Our goal is to determine which of the given statements about g ( x ) are true. We will analyze each statement individually, using the table of values to guide us.
Checking the first option First, let's check if g ( x ) = 4 ( x − 1 ) . We can plug in the x values from the table and see if we get the corresponding g ( x ) values. For x = 0 , g ( 0 ) = 4 ( 0 − 1 ) = 4 − 1 = 4 1 , but the table shows g ( 0 ) = 0 . Therefore, g ( x ) = 4 ( x − 1 ) .
Checking the second option Next, let's check if g ( x ) = 4 x − 1 . For x = 0 , g ( 0 ) = 4 0 − 1 = 1 − 1 = 0 . For x = 1 , g ( 1 ) = 4 1 − 1 = 4 − 1 = 3 . For x = 2 , g ( 2 ) = 4 2 − 1 = 16 − 1 = 15 . For x = 3 , g ( 3 ) = 4 3 − 1 = 64 − 1 = 63 . These values match the g ( x ) values in the table, so g ( x ) = 4 x − 1 is true.
Analyzing graph translations Now, let's consider the graph translations. Since g ( x ) = 4 x − 1 = f ( x ) − 1 , the graph of g ( x ) is the graph of f ( x ) translated down 1 unit.
Comparing domain and range Finally, let's compare the domain and range of f ( x ) and g ( x ) . The domain of f ( x ) = 4 x is all real numbers, and the range is ( 0 , ∞ ) . The domain of g ( x ) = 4 x − 1 is also all real numbers, but the range is ( − 1 , ∞ ) . Therefore, the domain and range of g ( x ) are not the same as the domain and range of f ( x ) .
Final Answer Based on our analysis, the two true statements are:
g ( x ) = 4 x − 1 The graph is translated down 1 unit.
Examples
Understanding function transformations is crucial in many fields, such as physics and engineering. For example, when analyzing the motion of a projectile, a vertical translation of the trajectory function can represent the effect of changing the initial height from which the projectile is launched. Similarly, in signal processing, transformations of functions are used to analyze and manipulate signals, such as audio or video signals. Knowing that g ( x ) = f ( x ) − 1 means that the graph of f ( x ) is shifted down by one unit helps us to visualize and understand the relationship between the two functions. This concept is widely used in computer graphics to manipulate objects in a scene, such as moving an object up or down by a certain amount.
By using the formula Q = I × t , we calculate the charge delivered by a current of 15.0 A over 30 seconds to be 450 C. Then, using the charge of a single electron, we find that approximately 2.81 quintillion electrons, or 2.81 × 1 0 21 , flow through the device. This process highlights the relationship between current, charge, and the flow of electrons.
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