$f(x)=x^2+1 \quad g(x)=5-x$
$(f+g)(x)=$
A. $x^2+x-4$
B. $x^2+x+4$
C. $x^2-x+6$
D. $x^2+x+6$
Answer (2)
Add the two functions: ( f + g ) ( x ) = f ( x ) + g ( x ) .
Substitute the expressions: ( f + g ) ( x ) = ( x 2 + 1 ) + ( 5 − x ) .
Simplify by combining like terms: ( f + g ) ( x ) = x 2 − x + 6 .
The final answer is x 2 − x + 6 .
Explanation
Understanding the Problem We are given two functions, f ( x ) = x 2 + 1 and g ( x ) = 5 − x . Our goal is to find the expression for ( f + g ) ( x ) , which means we need to add the two functions together.
Adding the Functions To find ( f + g ) ( x ) , we simply add f ( x ) and g ( x ) :
( f + g ) ( x ) = f ( x ) + g ( x ) Substituting the given expressions for f ( x ) and g ( x ) , we get: ( f + g ) ( x ) = ( x 2 + 1 ) + ( 5 − x ) Now, we simplify the expression by combining like terms.
Simplifying the Expression Combining the constant terms, we have 1 + 5 = 6 . So the expression becomes: ( f + g ) ( x ) = x 2 − x + 6 Thus, ( f + g ) ( x ) = x 2 − x + 6 .
Final Answer Therefore, ( f + g ) ( x ) = x 2 − x + 6 .
Examples
Understanding function addition is crucial in many real-world applications. For instance, if you're tracking the total cost of producing a product, f ( x ) might represent the fixed costs (like rent and equipment), and g ( x ) could represent the variable costs (like materials and labor) depending on the number of units x produced. Adding these functions, ( f + g ) ( x ) , gives you the total cost function, allowing you to analyze how costs change with production volume. This concept is also used in physics to combine different forces or movements and in computer graphics to combine transformations.
We found that ( f + g ) ( x ) = x 2 − x + 6 by adding the two functions. This matches option C from the provided choices. Thus, the answer to the question is option C.
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