Given that [tex]f(x)=2 x^3-x^2+5 x-2[/tex], find:
(a) [tex]F(0)[/tex]
(b) [tex]F(1)[/tex]
Answer (2)
Substitute x = 0 into f ( x ) to find f ( 0 ) .
Calculate f ( 0 ) = 2 ( 0 ) 3 − ( 0 ) 2 + 5 ( 0 ) − 2 = − 2 .
Substitute x = 1 into f ( x ) to find f ( 1 ) .
Calculate f ( 1 ) = 2 ( 1 ) 3 − ( 1 ) 2 + 5 ( 1 ) − 2 = 4 .
f ( 0 ) = − 2 , f ( 1 ) = 4
Explanation
Understanding the Problem We are given the function f ( x ) = 2 x 3 − x 2 + 5 x − 2 and asked to find the values of f ( 0 ) and f ( 1 ) . This involves substituting x = 0 and x = 1 into the expression for f ( x ) and evaluating the results.
Calculating f(0) To find f ( 0 ) , we substitute x = 0 into the expression for f ( x ) : f ( 0 ) = 2 ( 0 ) 3 − ( 0 ) 2 + 5 ( 0 ) − 2
Simplifying f(0) Now, we simplify the expression: f ( 0 ) = 2 ( 0 ) − 0 + 0 − 2 = 0 − 0 + 0 − 2 = − 2
Calculating f(1) To find f ( 1 ) , we substitute x = 1 into the expression for f ( x ) : f ( 1 ) = 2 ( 1 ) 3 − ( 1 ) 2 + 5 ( 1 ) − 2
Simplifying f(1) Now, we simplify the expression: f ( 1 ) = 2 ( 1 ) − 1 + 5 − 2 = 2 − 1 + 5 − 2 = 4
Final Answer Therefore, f ( 0 ) = − 2 and f ( 1 ) = 4 .
Examples
Evaluating functions is a fundamental concept in mathematics and has numerous real-world applications. For instance, in physics, a function might describe the position of an object at a given time. If f ( t ) = 3 t 2 + 2 t + 1 represents the distance an object has traveled after t seconds, evaluating f ( 0 ) would tell us the object's initial position, and f ( 1 ) would tell us its position after 1 second. Similarly, in economics, a cost function C ( x ) might represent the cost of producing x items. Evaluating C ( 0 ) would give the fixed costs, and C ( 1 ) would give the cost of producing one item.
Evaluating the function f ( x ) = 2 x 3 − x 2 + 5 x − 2 at x = 0 gives f ( 0 ) = − 2 and at x = 1 gives f ( 1 ) = 4 .
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