For [tex]f(x)=4 x+1[/tex] and [tex]g(x)=x^2-5[/tex], find [tex](f+g)(x)[/tex]
A. [tex]4 x^3-4[/tex]
B. [tex]x^2+4 x+6[/tex]
C. [tex]4 x^2-19[/tex]
D. [tex]x^2+4 x-4[/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 ) .
Simplify by combining like terms: ( f + g ) ( x ) = x 2 + 4 x − 4 .
The final answer 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 . Our goal is to find the sum of these two functions, which is denoted as ( f + g ) ( x ) . This means we need to add the expressions for f ( x ) and g ( x ) 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 ) Now, we substitute the given expressions for f ( x ) and g ( x ) :
( f + g ) ( x ) = ( 4 x + 1 ) + ( x 2 − 5 ) Next, we simplify the expression by combining like terms.
Simplifying the Expression We simplify the expression by combining like terms: ( f + g ) ( x ) = 4 x + 1 + x 2 − 5 Rearrange the terms to have the polynomial in standard form (decreasing order of exponents): ( f + g ) ( x ) = x 2 + 4 x + 1 − 5 Combine the constant terms: ( f + g ) ( x ) = x 2 + 4 x − 4
Choosing the Correct Option Now, we compare our result with the given options: A. 4 x 3 − 4 B. x 2 + 4 x + 6 C. 4 x 2 − 19 D. x 2 + 4 x − 4 Our result, x 2 + 4 x − 4 , matches option D.
Final Answer Therefore, ( f + g ) ( x ) = x 2 + 4 x − 4 . The correct answer is D.
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 the number of items sold, then the profit P ( x ) can be found by subtracting the cost from the revenue: P ( x ) = R ( x ) − C ( x ) . This is an example of combining functions to model a real-world situation. Similarly, if you have two different investment accounts, A ( t ) and B ( t ) , where t is time, the total value of your investments is T ( t ) = A ( t ) + B ( t ) , which is another example of adding functions.
To find ( f + g ) ( x ) , we combine the functions f ( x ) = 4 x + 1 and g ( x ) = x 2 − 5 . The simplified result is ( f + g ) ( x ) = x 2 + 4 x − 4 , matching option D.
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