If f(x) = x²-6x+8, what is the value of f(7) - f(3)?
A) -1 B) 0 C) 14 D) 15 E) 16
Answer (2)
Evaluate f ( 7 ) : Substitute x = 7 into f ( x ) = x 2 − 6 x + 8 to get f ( 7 ) = 15 .
Evaluate f ( 3 ) : Substitute x = 3 into f ( x ) = x 2 − 6 x + 8 to get f ( 3 ) = − 1 .
Calculate the difference: Subtract f ( 3 ) from f ( 7 ) to find f ( 7 ) − f ( 3 ) = 15 − ( − 1 ) .
Determine the final answer: f ( 7 ) − f ( 3 ) = 16 , so the answer is 16 .
Explanation
Understanding the Problem We are given the function f ( x ) = x 2 − 6 x + 8 and asked to find the value of f ( 7 ) − f ( 3 ) . This means we need to evaluate the function at x = 7 and x = 3 , and then subtract the second result from the first.
Calculating f(7) First, let's calculate f ( 7 ) . We substitute x = 7 into the function: f ( 7 ) = ( 7 ) 2 − 6 ( 7 ) + 8
f ( 7 ) = 49 − 42 + 8
f ( 7 ) = 7 + 8
f ( 7 ) = 15
Calculating f(3) Next, let's calculate f ( 3 ) . We substitute x = 3 into the function: f ( 3 ) = ( 3 ) 2 − 6 ( 3 ) + 8
f ( 3 ) = 9 − 18 + 8
f ( 3 ) = − 9 + 8
f ( 3 ) = − 1
Finding the Difference Now, we need to find the difference f ( 7 ) − f ( 3 ) : f ( 7 ) − f ( 3 ) = 15 − ( − 1 )
f ( 7 ) − f ( 3 ) = 15 + 1
f ( 7 ) − f ( 3 ) = 16
Final Answer Therefore, the value of f ( 7 ) − f ( 3 ) is 16.
Examples
Understanding functions and evaluating them at specific points is a fundamental concept in mathematics with many real-world applications. For instance, if f ( x ) represents the profit of a company as a function of the number of products sold ( x ), then finding f ( 7 ) − f ( 3 ) would tell us how much more profit the company makes when selling 7 products compared to selling 3 products. This kind of analysis is crucial for making informed business decisions.
The value of f ( 7 ) − f ( 3 ) for the function f ( x ) = x 2 − 6 x + 8 is 16 .
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