If [tex]f(x)=4 x^2+1[/tex] and [tex]g(x)=x^2-5[/tex], find [tex](f+g)(x)[/tex]
A. [tex]3 x^2+6[/tex]
B. [tex]3 x^2-4[/tex]
C. [tex]5 x^2-4[/tex]
D. [tex]5 x^2-6[/tex]
Answer (2)
Add the two functions: ( f + g ) ( x ) = f ( x ) + g ( x ) .
Substitute the expressions: ( f + g ) ( x ) = ( 4 x 2 + 1 ) + ( x 2 − 5 ) .
Combine like terms: ( f + g ) ( x ) = 5 x 2 − 4 .
The final answer is 5 x 2 − 4 .
Explanation
Understanding the problem We are given two functions, f ( x ) = 4 x 2 + 1 and g ( x ) = x 2 − 5 , and we want to find ( f + g ) ( x ) , which means we need to add the two functions together.
Adding the functions To find ( f + g ) ( x ) , we add the expressions for f ( x ) and g ( x ) :
( f + g ) ( x ) = f ( x ) + g ( x ) ( f + g ) ( x ) = ( 4 x 2 + 1 ) + ( x 2 − 5 )
Combining like terms Now, we combine like terms. We add the x 2 terms and the constant terms: ( f + g ) ( x ) = ( 4 x 2 + x 2 ) + ( 1 − 5 ) ( f + g ) ( x ) = 5 x 2 − 4
Final Answer So, ( f + g ) ( x ) = 5 x 2 − 4 . This matches option C.
Examples
Understanding how to combine functions is useful in many real-world scenarios. For example, if a company's revenue is modeled by one function and its costs by another, combining these functions can help determine the company's profit function. Similarly, in physics, if you have functions describing the position of two objects, adding these functions can describe the position of their center of mass.
By adding the functions f ( x ) = 4 x 2 + 1 and g ( x ) = x 2 − 5 , we find that ( f + g ) ( x ) = 5 x 2 − 4 . Therefore, the correct answer is option C. This involves combining like terms from both functions.
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