If [tex]f(x)=5 x^3-2[/tex] and [tex]g(x)-x+1[/tex], find [tex](f-g)(x)[/tex].
A. [tex]5 x^2-x-1[/tex]
B. [tex]5 x^4-2[/tex]
C. [tex]5 x^3+1[/tex]
D. [tex]5 x^3-x-3[/tex]
Answer (2)
Subtract g ( x ) from f ( x ) : ( f − g ) ( x ) = f ( x ) − g ( x ) .
Substitute the given expressions: ( f − g ) ( x ) = ( 5 x 3 − 2 ) − ( x + 1 ) .
Simplify the expression: ( f − g ) ( x ) = 5 x 3 − x − 3 .
The final answer is: 5 x 3 − x − 3 .
Explanation
Understanding the problem We are given two functions, f ( x ) = 5 x 3 − 2 and g ( x ) = x + 1 . We need to find ( f − g ) ( x ) , which means we need to subtract g ( x ) from f ( x ) .
Setting up the subtraction To find ( f − g ) ( x ) , we subtract the function g ( x ) from the function f ( x ) :
( f − g ) ( x ) = f ( x ) − g ( x ) Now, substitute the given expressions for f ( x ) and g ( x ) :
( f − g ) ( x ) = ( 5 x 3 − 2 ) − ( x + 1 )
Simplifying the expression Next, we simplify the expression by removing the parentheses and combining like terms: ( f − g ) ( x ) = 5 x 3 − 2 − x − 1 ( f − g ) ( x ) = 5 x 3 − x − 3
Finding the correct answer Finally, we compare the simplified expression with the given options to find the correct answer. The simplified expression is 5 x 3 − x − 3 , which matches option D.
Examples
Understanding function subtraction is useful in many real-world scenarios. For example, if you have a revenue function f ( x ) representing total income from selling x items and a cost function g ( x ) representing the total cost of producing x items, then ( f − g ) ( x ) gives you the profit function. This helps you determine how much profit you make after accounting for costs. Similarly, in physics, if f ( t ) represents the distance traveled by an object and g ( t ) represents the distance traveled by another object, then ( f − g ) ( t ) represents the difference in their positions at time t .
To find ( f − g ) ( x ) , we subtract g ( x ) from f ( x ) resulting in ( f − g ) ( x ) = 5 x 3 − x − 3 . This matches option D. Hence, the correct answer is option D.
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