For [tex]f(x)=4 x+1[/tex] and [tex]g(x)=x^2-5[/tex], find [tex](f+g)(x)[/tex].
A. [tex]x^2+4 x-4[/tex]
B. [tex]x^2+4 x+6[/tex]
C. [tex]4 x^3-4[/tex]
D. [tex]4 x^2-19[/tex]
Answer (2)
Add the two functions: ( f + g ) ( x ) = f ( x ) + g ( x ) .
Substitute the expressions: ( f + g ) ( x ) = ( 4 x + 1 ) + ( x 2 − 5 ) .
Combine like terms: ( f + g ) ( x ) = x 2 + 4 x − 4 .
The result is: x 2 + 4 x − 4
Explanation
Understanding the Problem We are given two functions, f ( x ) = 4 x + 1 and g ( x ) = x 2 − 5 , and we want to find their sum, ( f + g ) ( x ) . This 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 + 1 ) + ( x 2 − 5 ) Now, we combine like terms.
Simplifying the Expression Combining like terms, we have: ( f + g ) ( x ) = x 2 + 4 x + ( 1 − 5 ) ( f + g ) ( x ) = x 2 + 4 x − 4
Final Answer Therefore, ( f + g ) ( x ) = x 2 + 4 x − 4 . This corresponds to option A.
Examples
Understanding how to combine functions is useful in many real-world scenarios. For example, if you have a business where your revenue R ( x ) depends on the number of items sold x , and your cost C ( x ) also depends on x , then the profit function P ( x ) is the difference between the revenue and the cost, i.e., P ( x ) = R ( x ) − C ( x ) . Combining functions allows you to model and analyze complex relationships in economics, physics, and engineering.
The combined function ( f + g ) ( x ) for f ( x ) = 4 x + 1 and g ( x ) = x 2 − 5 is x 2 + 4 x − 4 , which corresponds to option A.
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