If [tex]f(x)=x^2-2 x[/tex] and [tex]g(x)=6 x+4[/tex], for which value of [tex]x[/tex] does [tex](f+g)(x)=0[/tex]?
Answer (2)
First, find the expression for ( f + g ) ( x ) by adding f ( x ) and g ( x ) : ( f + g ) ( x ) = ( x 2 − 2 x ) + ( 6 x + 4 ) .
Simplify the expression: ( f + g ) ( x ) = x 2 + 4 x + 4 .
Set the simplified expression equal to 0: x 2 + 4 x + 4 = 0 .
Solve the quadratic equation by factoring: ( x + 2 ) ( x + 2 ) = 0 , which gives x = − 2 . Therefore, the final answer is − 2 .
Explanation
Understanding the Problem We are given two functions, f ( x ) = x 2 − 2 x and g ( x ) = 6 x + 4 . We want to find the value of x such that ( f + g ) ( x ) = 0 . This means we need to add the two functions together and then solve for x .
Adding the Functions First, let's find the expression for ( f + g ) ( x ) . We have ( f + g ) ( x ) = f ( x ) + g ( x ) Substituting the given expressions for f ( x ) and g ( x ) , we get ( f + g ) ( x ) = ( x 2 − 2 x ) + ( 6 x + 4 )
Simplifying the Expression Now, let's simplify the expression for ( f + g ) ( x ) by combining like terms: ( f + g ) ( x ) = x 2 − 2 x + 6 x + 4 = x 2 + 4 x + 4
Setting the Expression to Zero We want to find the value of x such that ( f + g ) ( x ) = 0 . So, we set the simplified expression equal to 0: x 2 + 4 x + 4 = 0
Solving the Quadratic Equation Now, we need to solve the quadratic equation x 2 + 4 x + 4 = 0 for x . We can factor the quadratic expression as follows: ( x + 2 ) ( x + 2 ) = 0 This can also be written as ( x + 2 ) 2 = 0
Finding the Value of x To find the value of x , we set the factor equal to 0: x + 2 = 0 Solving for x , we get x = − 2
Checking the Answer Therefore, the value of x for which ( f + g ) ( x ) = 0 is x = − 2 .
We can check our answer by plugging x = − 2 into the expression for ( f + g ) ( x ) :
( f + g ) ( − 2 ) = ( − 2 ) 2 + 4 ( − 2 ) + 4 = 4 − 8 + 4 = 0
Final Answer The value of x for which ( f + g ) ( x ) = 0 is − 2 .
Examples
Understanding functions and their combinations is crucial in many real-world applications. For instance, imagine a business where f ( x ) represents the cost of producing x items, and g ( x ) represents the revenue generated from selling x items. The function ( f + g ) ( x ) would then represent the overall profit (or loss) of the business. Finding the value of x where ( f + g ) ( x ) = 0 would tell the business owner the break-even point, where the costs equal the revenue. This kind of analysis helps in making informed decisions about production levels and pricing strategies.
The value of x for which (f+g)(x)=0 is -2, found by combining the functions f(x) and g(x) and simplifying the resulting equation. After setting the quadratic equation to zero and factoring, we determined that x must equal -2. A check confirms that this solution is accurate.
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