Select the correct answer.

Function [tex]$h$[/tex] is the product of functions [tex]$f$[/tex] and [tex]$g$[/tex].
[tex]
\begin{array}{l}
f(x)=2 x+5 \
g(x)=6 x-9
\end{array}
[/tex]
Which equation defines function [tex]$h$[/tex]?
A. [tex]$h(x)=12 x^2-4 x-45$[/tex]
B. [tex]$h(x)=12 x^2-45$[/tex]
C. [tex]$h(x)=12 x-45$[/tex]
D. [tex]$h(x)=12 x^2+12 x-45$[/tex]

Multiply the two functions: h ( x ) = f ( x ) × g ( x ) = ( 2 x + 5 ) ( 6 x − 9 ) .
Expand the product: h ( x ) = 12 x 2 − 18 x + 30 x − 45 .
Combine like terms: h ( x ) = 12 x 2 + 12 x − 45 .
The equation that defines function h is h ( x ) = 12 x 2 + 12 x − 45 ​ .

Explanation

Understanding the Problem We are given two functions, f ( x ) = 2 x + 5 and g ( x ) = 6 x − 9 , and we are told that h ( x ) is the product of f ( x ) and g ( x ) . Our goal is to find the equation that defines h ( x ) . This means we need to multiply the two functions together and simplify the resulting expression.

Multiplying the Functions To find h ( x ) , we multiply f ( x ) and g ( x ) :
h ( x ) = f ( x ) × g ( x ) = ( 2 x + 5 ) ( 6 x − 9 )

Expanding the Product Now, we expand the product using the distributive property (also known as the FOIL method): h ( x ) = ( 2 x ) ( 6 x ) + ( 2 x ) ( − 9 ) + ( 5 ) ( 6 x ) + ( 5 ) ( − 9 )

Performing Multiplications Next, we perform the multiplications: h ( x ) = 12 x 2 − 18 x + 30 x − 45

Simplifying the Expression Finally, we combine like terms to simplify the expression: h ( x ) = 12 x 2 + ( − 18 x + 30 x ) − 45 h ( x ) = 12 x 2 + 12 x − 45

Finding the Correct Option So, the equation that defines function h is h ( x ) = 12 x 2 + 12 x − 45 . Comparing this to the given options, we see that it matches option D.


Examples
Understanding how functions combine is crucial in many real-world applications. For instance, if you're running a business, you might have a cost function f ( x ) that represents the cost of producing x items and a revenue function g ( x ) that represents the revenue from selling x items. The profit function h ( x ) , which is the product of a price function and the number of items sold, can be analyzed to optimize production levels and maximize profits. Similarly, in physics, the combination of different forces or motions can be described by combining functions.

Answered by GinnyAnswer | 2025-07-03

The function h ( x ) , defined as the product of f ( x ) and g ( x ) , is calculated to be h ( x ) = 12 x 2 + 12 x − 45 . The correct option is D. This was determined by expanding and simplifying the product of the two given linear functions.
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Answered by Anonymous | 2025-07-04