If [tex]f(x)=5 x^3-2[/tex] and [tex]g(x)=2 x+1[/tex], find [tex](f+g)(x)[/tex].
A. [tex]10 x^3-2[/tex]
B. [tex]5 x^3+2 x-1[/tex]
C. [tex]3 x^3-1[/tex]
D. [tex]3 x^2-1[/tex]
Answer (2)
Add the two functions: ( f + g ) ( x ) = f ( x ) + g ( x ) .
Substitute the given expressions: ( f + g ) ( x ) = ( 5 x 3 − 2 ) + ( 2 x + 1 ) .
Simplify the expression: ( f + g ) ( x ) = 5 x 3 + 2 x − 1 .
The final answer is: 5 x 3 + 2 x − 1 .
Explanation
Understanding the Problem We are given two functions, f ( x ) = 5 x 3 − 2 and g ( x ) = 2 x + 1 . Our goal is to find the sum of these two functions, which is denoted as ( f + g ) ( x ) . This simply 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 two functions: ( f + g ) ( x ) = f ( x ) + g ( x ) Now, substitute the given expressions for f ( x ) and g ( x ) :
( f + g ) ( x ) = ( 5 x 3 − 2 ) + ( 2 x + 1 ) Next, we simplify the expression by combining like terms.
Simplifying the Expression Combine the constant terms: − 2 + 1 = − 1 . So we have: ( f + g ) ( x ) = 5 x 3 + 2 x − 1 Now, we compare this simplified expression with the given options to find the correct answer.
Finding the Correct Option The simplified expression is 5 x 3 + 2 x − 1 , which matches option B. Therefore, ( f + g ) ( x ) = 5 x 3 + 2 x − 1 .
Examples
Understanding function addition is useful in many real-world scenarios. For example, if a company's revenue, f ( x ) , and expenses, g ( x ) , are modeled as functions of time, x , then the profit, ( f + g ) ( x ) , can be found by adding the revenue and expense functions. This allows the company to easily analyze how profit changes over time. Similarly, in physics, if you have two forces acting on an object, you can add their force vectors (which are functions of position) to find the net force acting on the object.
The sum of the functions is calculated as ( f + g ) ( x ) = ( 5 x 3 − 2 ) + ( 2 x + 1 ) , which simplifies to 5 x 3 + 2 x − 1 . The correct answer is option B: 5 x 3 + 2 x − 1 .
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