If [tex]f(x)=2x[/tex] and [tex]g(x)=5x+2[/tex], what is [tex](f \cdot g)(x)[/tex] when [tex]x=3[/tex]?

First, find the product of the functions: ( f \t \tcdot g ) ( x ) = f ( x ) \t \tcdot g ( x ) = ( 2 x ) ( 5 x + 2 ) = 10 x 2 + 4 x .
Then, substitute x = 3 into the expression: ( f \t \tcdot g ) ( 3 ) = 10 ( 3 ) 2 + 4 ( 3 ) .
Calculate the value: 10 ( 3 ) 2 + 4 ( 3 ) = 10 ( 9 ) + 12 = 90 + 12 = 102 .
The final answer is: 102 ​ .

Explanation

Understanding the Problem We are given two functions, f ( x ) = 2 x and g ( x ) = 5 x + 2 . We need to find the value of ( f \t \tcdot g ) ( x ) when x = 3 . The notation ( f \t \tcdot g ) ( x ) means the product of the two functions, i.e., f ( x ) \t \tcdot g ( x ) .

Finding the Product of the Functions First, we find the expression for ( f \t \tcdot g ) ( x ) by multiplying f ( x ) and g ( x ) : ( f \t \tcdot g ) ( x ) = f ( x ) \t \tcdot g ( x ) = ( 2 x ) ( 5 x + 2 )

Simplifying the Expression Next, we simplify the expression ( 2 x ) ( 5 x + 2 ) : ( 2 x ) ( 5 x + 2 ) = 10 x 2 + 4 x So, ( f \t \tcdot g ) ( x ) = 10 x 2 + 4 x .

Substituting x = 3 Now, we substitute x = 3 into the expression for ( f \t \tcdot g ) ( x ) : ( f \t \tcdot g ) ( 3 ) = 10 ( 3 ) 2 + 4 ( 3 )

Calculating the Final Value Finally, we calculate the value of 10 ( 3 ) 2 + 4 ( 3 ) : 10 ( 3 ) 2 + 4 ( 3 ) = 10 ( 9 ) + 12 = 90 + 12 = 102 Therefore, ( f \t \tcdot g ) ( 3 ) = 102 .


Examples
Understanding function operations like multiplication is crucial in many real-world applications. For instance, if f ( x ) represents the number of items produced per worker and g ( x ) represents the revenue per item, then ( f \t \tcdot g ) ( x ) gives the total revenue generated by x workers. Evaluating this at a specific value, like x = 3 , helps businesses estimate their total revenue based on the number of workers. This concept is also used in physics to calculate power, where power is the product of voltage and current.

Answered by GinnyAnswer | 2025-07-05