An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (2)
The functions are: ( f + g ) ( x ) = x + 6 x + 4 + x − 8 , ( f − g ) ( x ) = x + 6 x + 4 − x − 8 , ( f g ) ( x ) = x + 6 ( x + 4 ) ( x − 8 ) , and ( g f ) ( x ) = ( x + 6 ) ( x − 8 ) x + 4 .
The domain of f + g , f − g , and f g is [ 8 , ∞ ) .
The domain of g f is ( 8 , ∞ ) .
The final answer is: ( 8 , ∞ )
Explanation
Problem Analysis We are given two functions, $f(x) =
\sqrt{\frac{x+4}{x+6}} an d g(x) = \sqrt{x-8} , an d w e n ee d t o f in d t h e f u n c t i o n s f+g, f-g, fg, \frac{f}{g}$ and their domains.
Finding the Combined Functions First, let's find the expressions for the combined functions:
( f + g ) ( x ) = f ( x ) + g ( x ) = x + 6 x + 4 + x − 8
( f − g ) ( x ) = f ( x ) − g ( x ) = x + 6 x + 4 − x − 8
( f g ) ( x ) = f ( x ) g ( x ) = x + 6 x + 4 ⋅ x − 8 = x + 6 ( x + 4 ) ( x − 8 )
( g f ) ( x ) = g ( x ) f ( x ) = x − 8 x + 6 x + 4 = ( x + 6 ) ( x − 8 ) x + 4
Finding the Domains Now, let's determine the domains of these functions. The domain of f ( x ) is determined by the conditions x + 6 x + 4 ≥ 0 and x = − 6 . The domain of g ( x ) is determined by the condition x − 8 ≥ 0 , which means x ≥ 8 .
For f + g , f − g , and f g , we need both f ( x ) and g ( x ) to be defined. Thus, we need x + 6 x + 4 ≥ 0 and x ≥ 8 . Since x ≥ 8 , we have 0"> x + 4 > 0 and 0"> x + 6 > 0 , so 0"> x + 6 x + 4 > 0 . Thus, the domain of f + g , f − g , and f g is [ 8 , ∞ ) .
For g f , we need ( x + 6 ) ( x − 8 ) x + 4 ≥ 0 and g ( x ) = 0 , which means 0"> x − 8 > 0 , so 8"> x > 8 . Thus, we need to solve the inequality ( x + 6 ) ( x − 8 ) x + 4 ≥ 0 with the condition 8"> x > 8 . If 8"> x > 8 , then 0, x+6 > 0, x-8 > 0"> x + 4 > 0 , x + 6 > 0 , x − 8 > 0 , so the fraction is positive. Thus, the domain of g f is ( 8 , ∞ ) .
Final Answer Therefore, the domain of g f is ( 8 , ∞ ) .
Examples
Understanding function composition and domain restrictions is crucial in many fields, such as physics and engineering. For example, when analyzing the motion of an object, one function might describe its position over time, while another describes the force acting upon it. Combining these functions allows engineers to predict the object's behavior under various conditions, but they must also consider the domain restrictions to ensure the functions are physically meaningful. For instance, a function representing the square root of time would only be valid for non-negative time values, reflecting the real-world constraint that time cannot be negative.
Approximately 2.81 billion billion electrons flow through the device that delivers a current of 15.0 A for 30 seconds. The total charge is calculated using the formula Q = I × t, yielding 450 C. This charge is then divided by the charge of a single electron to find the number of electrons.
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