If [tex]f(x)=4-x^2[/tex] and [tex]g(x)=6x[/tex], which expression is equivalent to [tex](g-f)(3)[/tex]?
A. [tex]6-3-(4+3)^2[/tex]
B. [tex]6-3-(4-3^2)[/tex]
C. [tex]6(3)-4+3^2[/tex]
D. [tex]6(3)-4-3^2[/tex]
Answer (2)
First, find the expression for ( g − f ) ( x ) by subtracting f ( x ) from g ( x ) : ( g − f ) ( x ) = 6 x − ( 4 − x 2 ) = 6 x − 4 + x 2 .
Substitute x = 3 into the expression for ( g − f ) ( x ) : ( g − f ) ( 3 ) = 6 ( 3 ) − 4 + ( 3 ) 2 = 18 − 4 + 9 = 23 .
Compare the result with the given options to find the equivalent expression.
The expression equivalent to ( g − f ) ( 3 ) is 6 ( 3 ) − 4 + 3 2 .
Explanation
Understanding the problem We are given two functions, f ( x ) = 4 − x 2 and g ( x ) = 6 x . We want to find an expression that is equivalent to ( g − f ) ( 3 ) . First, we need to find the expression for ( g − f ) ( x ) .
Finding (g-f)(x) The expression ( g − f ) ( x ) means g ( x ) − f ( x ) . So, we have ( g − f ) ( x ) = g ( x ) − f ( x ) = 6 x − ( 4 − x 2 ) = 6 x − 4 + x 2 .
Evaluating (g-f)(3) Now, we need to evaluate ( g − f ) ( 3 ) . We substitute x = 3 into the expression we found for ( g − f ) ( x ) :
( g − f ) ( 3 ) = 6 ( 3 ) − 4 + ( 3 ) 2 = 18 − 4 + 9 = 14 + 9 = 23.
Comparing with the options Now we need to find which of the given options is equal to 23. Option 1: 6 − 3 − ( 4 + 3 ) 2 = 3 − ( 7 ) 2 = 3 − 49 = − 46 Option 2: 6 − 3 − ( 4 − 3 2 ) = 3 − ( 4 − 9 ) = 3 − ( − 5 ) = 3 + 5 = 8 Option 3: 6 ( 3 ) − 4 + 3 2 = 18 − 4 + 9 = 14 + 9 = 23 Option 4: 6 ( 3 ) − 4 − 3 2 = 18 − 4 − 9 = 14 − 9 = 5 Only option 3 is equal to 23.
Final Answer Therefore, the expression equivalent to ( g − f ) ( 3 ) is 6 ( 3 ) − 4 + 3 2 .
Examples
Understanding function operations like ( g − f ) ( x ) is useful in many real-world scenarios. For example, imagine g ( x ) represents the revenue from selling x items, and f ( x ) represents the cost of producing x items. Then ( g − f ) ( x ) represents the profit from selling x items. Evaluating ( g − f ) ( 3 ) would tell you the profit from selling 3 items. This kind of analysis helps businesses make informed decisions about production and pricing.
The expression equivalent to ( g − f ) ( 3 ) is found to be 6 ( 3 ) − 4 + 3 2 , which equals 23. This corresponds to Option C from the provided choices. We confirm that this option yields the correct result upon evaluation.
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