$f(x)=x+2 \quad g(x)=x-4$
$(f g)(x)=$
$2 x-2$
$x^2-8$
$x^2-2 x-8$
$x^2+2 x-8$
Answer (2)
Multiply the two functions: ( f g ) ( x ) = ( x + 2 ) ( x − 4 ) .
Expand the product: ( x + 2 ) ( x − 4 ) = x 2 − 4 x + 2 x − 8 .
Simplify the expression: x 2 − 2 x − 8 .
The product of the functions is x 2 − 2 x − 8 .
Explanation
Understanding the Problem We are given two functions, f ( x ) = x + 2 and g ( x ) = x − 4 . We need to find the product of these two functions, which is denoted as ( f g ) ( x ) or f ( x ) g ( x ) . This means we need to multiply the expressions for f ( x ) and g ( x ) .
Multiplying the Functions To find the product of the functions, we multiply their expressions: ( f g ) ( x ) = f ( x ) ⋅ g ( x ) = ( x + 2 ) ( x − 4 ) Now, we expand this product using the distributive property (also known as the FOIL method):
Expanding the Product Expanding the product ( x + 2 ) ( x − 4 ) , we get: x ( x − 4 ) + 2 ( x − 4 ) = x 2 − 4 x + 2 x − 8 Now, we simplify the expression by combining like terms:
Simplifying the Expression Combining the like terms − 4 x and + 2 x , we have: x 2 − 4 x + 2 x − 8 = x 2 − 2 x − 8 So, the product of the functions is x 2 − 2 x − 8 .
Final Answer Comparing our result with the given options, we see that the correct answer is x 2 − 2 x − 8 .
Therefore, ( f g ) ( x ) = x 2 − 2 x − 8 .
Examples
Understanding function multiplication is crucial in many real-world applications. For instance, if you're calculating the area of a rectangle where the length is defined by f ( x ) = x + 2 and the width is defined by g ( x ) = x − 4 , the area would be ( f g ) ( x ) = x 2 − 2 x − 8 . This concept extends to more complex scenarios, such as modeling the growth of populations or analyzing the behavior of systems in engineering.
The product of the functions defined by f ( x ) = x + 2 and g ( x ) = x − 4 is found by multiplying them together and simplifying the result. The final expression is ( f g ) ( x ) = x 2 − 2 x − 8 . Therefore, the correct answer is x 2 − 2 x − 8 .
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