If [tex]f(x)=2 x^2+3[/tex] and [tex]g(x)=x^2-7[/tex], find [tex](f-g)(x)[/tex]
A. [tex]3 x^2-10[/tex]
B. [tex]x^2-4[/tex]
C. [tex]x^2+10[/tex]
D. [tex]3 x^2-4[/tex]
Answer (2)
Subtract g ( x ) from f ( x ) : ( f − g ) ( x ) = f ( x ) − g ( x ) .
Substitute the expressions: ( f − g ) ( x ) = ( 2 x 2 + 3 ) − ( x 2 − 7 ) .
Simplify the expression: ( f − g ) ( x ) = 2 x 2 + 3 − x 2 + 7 .
Combine like terms to find the final answer: ( f − g ) ( x ) = x 2 + 10 , so the answer is x 2 + 10 .
Explanation
Understanding the problem We are given two functions, f ( x ) = 2 x 2 + 3 and g ( x ) = x 2 − 7 , and we need to find ( f − g ) ( x ) . This means we need to subtract the function g ( x ) from the function f ( x ) .
Setting up the subtraction To find ( f − g ) ( x ) , we subtract g ( x ) from f ( x ) : ( f − g ) ( x ) = f ( x ) − g ( x ) Now, substitute the given expressions for f ( x ) and g ( x ) : ( f − g ) ( x ) = ( 2 x 2 + 3 ) − ( x 2 − 7 )
Simplifying the expression Next, we simplify the expression by distributing the negative sign and combining like terms: ( f − g ) ( x ) = 2 x 2 + 3 − x 2 + 7 Combine the x 2 terms and the constant terms: ( f − g ) ( x ) = ( 2 x 2 − x 2 ) + ( 3 + 7 )
Final result Finally, we simplify further to get the expression for ( f − g ) ( x ) :
( f − g ) ( x ) = x 2 + 10 So, ( f − g ) ( x ) = x 2 + 10 .
Examples
Understanding function operations like subtraction is crucial in many real-world applications. For example, if f ( x ) represents the total revenue of a company and g ( x ) represents the total cost, then ( f − g ) ( x ) gives the profit. By analyzing the profit function, the company can make informed decisions about pricing, production, and cost management. Similarly, in physics, if f ( x ) represents the position of an object at time x and g ( x ) represents the position of another object, then ( f − g ) ( x ) represents the relative distance between the two objects.
After subtracting the function g ( x ) from f ( x ) , we find that ( f − g ) ( x ) = x 2 + 10 . Therefore, the correct answer is option C: x 2 + 10 . This involved simple algebraic steps of subtraction and combining like terms.
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