An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (2)
To verify that g ( x ) is the inverse of f ( x ) , we need to check if f ( g ( x )) = x and g ( f ( x )) = x .
Calculate f ( g ( x )) = f ( 3 1 x ) = 3 ( 3 1 x ) = x .
Calculate g ( f ( x )) = g ( 3 x ) = 3 1 ( 3 x ) = x .
The expression that verifies the inverse relationship is 3 1 ( 3 x ) .
3 1 ( 3 x )
Explanation
Understanding the Problem We are given two functions, f ( x ) = 3 x and g ( x ) = 3 1 x . We want to determine which expression can be used to verify that g ( x ) is the inverse of f ( x ) .
Verifying Inverse Functions To verify that g ( x ) is the inverse of f ( x ) , we need to check if f ( g ( x )) = x and g ( f ( x )) = x . This means that when we compose the two functions in either order, we should get the identity function, which is just x .
Calculating f(g(x)) Let's compute f ( g ( x )) . We substitute g ( x ) into f ( x ) : f ( g ( x )) = f ( 3 1 x ) = 3 ( 3 1 x ) = x This shows that f ( g ( x )) = x .
Calculating g(f(x)) Now let's compute g ( f ( x )) . We substitute f ( x ) into g ( x ) :
g ( f ( x )) = g ( 3 x ) = 3 1 ( 3 x ) = x This shows that g ( f ( x )) = x .
Identifying the Correct Expression Both f ( g ( x )) and g ( f ( x )) equal x , so g ( x ) is indeed the inverse of f ( x ) . The expressions that could be used to verify this are 3 ( 3 1 x ) and 3 1 ( 3 x ) . Looking at the given options, we see that 3 1 ( 3 x ) is one of the choices.
Final Answer Therefore, the expression that could be used to verify that g ( x ) is the inverse of f ( x ) is 3 1 ( 3 x ) .
Examples
Understanding inverse functions is crucial in many real-world applications. For example, converting temperatures between Celsius and Fahrenheit involves inverse functions. If f ( x ) converts Celsius to Fahrenheit, then its inverse g ( x ) converts Fahrenheit back to Celsius. Verifying that these functions are indeed inverses ensures accurate conversions, which is vital in fields like meteorology, engineering, and medicine where precise temperature measurements are essential.
The total charge that flows through the electric device is 450 coulombs. This means that approximately 2.81 billion billion electrons flow through the device while delivering a current of 15.0 A for 30 seconds. We calculated this using the current, time, and the charge of a single electron.
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