If [tex]f(x)=3 x^2+1[/tex] and [tex]g(x)=1-x[/tex], what is the value of [tex](f-g)(2)[/tex]?
Answer (1)
Evaluate f ( 2 ) by substituting x = 2 into f ( x ) = 3 x 2 + 1 , which gives f ( 2 ) = 13 .
Evaluate g ( 2 ) by substituting x = 2 into g ( x ) = 1 − x , which gives g ( 2 ) = − 1 .
Calculate ( f − g ) ( 2 ) by subtracting g ( 2 ) from f ( 2 ) , so ( f − g ) ( 2 ) = 13 − ( − 1 ) .
Simplify the result to obtain the final answer: ( f − g ) ( 2 ) = 14 , so the final answer is 14 .
Explanation
Understanding the Problem We are given two functions, f ( x ) = 3 x 2 + 1 and g ( x ) = 1 − x . Our goal is to find the value of ( f − g ) ( 2 ) . Remember that ( f − g ) ( x ) means f ( x ) − g ( x ) . So, we need to calculate f ( 2 ) and g ( 2 ) and then subtract g ( 2 ) from f ( 2 ) .
Calculating f(2) First, let's find f ( 2 ) . We substitute x = 2 into the expression for f ( x ) : f ( 2 ) = 3 ( 2 ) 2 + 1 = 3 ( 4 ) + 1 = 12 + 1 = 13.
Calculating g(2) Next, let's find g ( 2 ) . We substitute x = 2 into the expression for g ( x ) : g ( 2 ) = 1 − 2 = − 1.
Calculating (f-g)(2) Now, we can find ( f − g ) ( 2 ) by subtracting g ( 2 ) from f ( 2 ) : ( f − g ) ( 2 ) = f ( 2 ) − g ( 2 ) = 13 − ( − 1 ) = 13 + 1 = 14.
Final Answer Therefore, the value of ( f − g ) ( 2 ) is 14.
Examples
Understanding function operations like ( f − g ) ( x ) is useful in many real-world scenarios. For example, imagine f ( x ) represents the revenue of a company from selling x units of a product, and g ( x ) represents the cost of producing x units. Then, ( f − g ) ( x ) would represent the profit the company makes from selling x units. Evaluating ( f − g ) ( 2 ) would tell you the profit made from selling 2 units. This kind of analysis helps businesses make informed decisions about production and pricing.
