An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (1)
The function is f ( x ) = − x , and its domain is x ≥ 0 .
Check each point in the given options to see if it satisfies the equation f ( x ) = − x .
Option 1, 2, and 3 have points that do not satisfy the equation or have negative x values.
Option 4: ( 1 , − 1 ) , ( 4 , − 2 ) , ( 9 , − 3 ) satisfies the equation. Therefore, the answer is ( 1 , − 1 ) , ( 4 , − 2 ) , ( 9 , − 3 ) .
Explanation
Understanding the Problem We are given the function f ( x ) = − x and four lists of points. We need to determine which list contains three points that all lie on the graph of the function. This means that for each point ( x , y ) in the correct list, the equation y = − x must be true. Also, since we are taking the square root of x , x must be non-negative ( x ≥ 0 ).
Analyzing Option 1 Let's analyze each option:
Option 1: ( − 9 , 3 ) , ( − 4 , 2 ) , ( − 1 , 1 ) Since the domain of the function is x ≥ 0 , we can immediately eliminate this option because all the x -values are negative.
Analyzing Option 2 Option 2: ( 1 , 1 ) , ( 4 , 2 ) , ( 9 , 3 ) Let's check each point:
For ( 1 , 1 ) : f ( 1 ) = − 1 = − 1 . Since − 1 = 1 , this point is not on the graph.
For ( 4 , 2 ) : f ( 4 ) = − 4 = − 2 . Since − 2 = 2 , this point is not on the graph.
For ( 9 , 3 ) : f ( 9 ) = − 9 = − 3 . Since − 3 = 3 , this point is not on the graph. Therefore, this option is incorrect.
Analyzing Option 3 Option 3: ( − 9 , − 3 ) , ( − 4 , − 2 ) , ( − 1 , − 1 ) Again, since the domain of the function is x ≥ 0 , we can immediately eliminate this option because all the x -values are negative.
Analyzing Option 4 Option 4: ( 1 , − 1 ) , ( 4 , − 2 ) , ( 9 , − 3 ) Let's check each point:
For ( 1 , − 1 ) : f ( 1 ) = − 1 = − 1 . This point is on the graph.
For ( 4 , − 2 ) : f ( 4 ) = − 4 = − 2 . This point is on the graph.
For ( 9 , − 3 ) : f ( 9 ) = − 9 = − 3 . This point is on the graph. Therefore, this option is correct.
Final Answer The list of points that lie on the graph of the function f ( x ) = − x is ( 1 , − 1 ) , ( 4 , − 2 ) , ( 9 , − 3 ) .
Examples
Understanding functions and their graphs is crucial in many real-world applications. For example, engineers use functions to model the behavior of physical systems, such as the trajectory of a projectile or the flow of electricity in a circuit. By analyzing the graph of a function, engineers can gain insights into the system's behavior and make predictions about its future performance. Similarly, economists use functions to model economic phenomena, such as the relationship between supply and demand or the growth of a country's GDP. By understanding the properties of these functions, economists can make informed decisions about economic policy.
