For the function [tex]f(x)=\frac{7 x^2-3}{x^2}[/tex] find the values of [tex]f(10), f(50), f(100)[/tex], and [tex]f(1000)[/tex].
Answer (2)
Calculate f ( 10 ) : f ( 10 ) = 1 0 2 7 ( 10 ) 2 − 3 = 6.97 .
Calculate f ( 50 ) : f ( 50 ) = 5 0 2 7 ( 50 ) 2 − 3 = 6.9988 .
Calculate f ( 100 ) : f ( 100 ) = 10 0 2 7 ( 100 ) 2 − 3 = 6.9997 .
Calculate f ( 1000 ) : f ( 1000 ) = 100 0 2 7 ( 1000 ) 2 − 3 = 6.999997 . The correct answer is a.
Explanation
Understanding the Problem We are given the function f ( x ) = x 2 7 x 2 − 3 and asked to find the values of f ( 10 ) , f ( 50 ) , f ( 100 ) , and f ( 1000 ) . This involves substituting these values into the function and calculating the results.
Calculating f(10) First, let's calculate f ( 10 ) :
f ( 10 ) = ( 10 ) 2 7 ( 10 ) 2 − 3 = 100 7 ( 100 ) − 3 = 100 700 − 3 = 100 697 = 6.97
Calculating f(50) Next, let's calculate f ( 50 ) :
f ( 50 ) = ( 50 ) 2 7 ( 50 ) 2 − 3 = 2500 7 ( 2500 ) − 3 = 2500 17500 − 3 = 2500 17497 = 6.9988
Calculating f(100) Now, let's calculate f ( 100 ) :
f ( 100 ) = ( 100 ) 2 7 ( 100 ) 2 − 3 = 10000 7 ( 10000 ) − 3 = 10000 70000 − 3 = 10000 69997 = 6.9997
Calculating f(1000) Finally, let's calculate f ( 1000 ) :
f ( 1000 ) = ( 1000 ) 2 7 ( 1000 ) 2 − 3 = 1000000 7 ( 1000000 ) − 3 = 1000000 7000000 − 3 = 1000000 6999997 = 6.999997
Selecting the Correct Answer Comparing our calculated values with the multiple choice options, we find that option a matches our results: f ( 10 ) = 6.97 f ( 50 ) = 6.9988 f ( 100 ) = 6.9997 f ( 1000 ) = 6.999997
Examples
Understanding functions and their evaluations is crucial in many real-world applications. For instance, in physics, you might use a function to model the trajectory of a projectile. By plugging in different time values, you can determine the projectile's position at those times. Similarly, in economics, functions can represent cost curves or revenue streams, and evaluating them at different production levels helps in making informed decisions about resource allocation and pricing strategies. This exercise reinforces the fundamental skill of function evaluation, which is a building block for more complex mathematical modeling.
The values calculated for the function f ( x ) = x 2 7 x 2 − 3 are as follows: f ( 10 ) = 6.97 , f ( 50 ) = 6.9988 , f ( 100 ) = 6.9997 , and f ( 1000 ) = 6.999997 .
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