Study the equations:
[tex]
\begin{array}{l}
f(x)=6 x+7 \\
g(x)=4 x-2
\end{array}
[/tex]
What is [tex]h(x)=f(x) g(x)[/tex] ?
A. [tex]h(x)=24 x^2-14[/tex]
B. [tex]h(x)=24 x^2+40 x+14[/tex]
C. [tex]h(x)=24 x^2+16 x-14[/tex]
D. [tex]h(x)=24 x^2+12 x-14[/tex]
Answer (1)
Multiply the two functions: h ( x ) = ( 6 x + 7 ) ( 4 x − 2 ) .
Expand the product using the distributive property: h ( x ) = 24 x 2 − 12 x + 28 x − 14 .
Combine like terms: h ( x ) = 24 x 2 + 16 x − 14 .
The final answer is 24 x 2 + 16 x − 14 .
Explanation
Understanding the Problem We are given two functions, f ( x ) = 6 x + 7 and g ( x ) = 4 x − 2 , and we want to find their product, h ( x ) = f ( x ) g ( x ) . This means we need to multiply the two expressions together.
Multiplying the Functions To find h ( x ) , we multiply f ( x ) and g ( x ) :
h ( x ) = f ( x ) g ( x ) = ( 6 x + 7 ) ( 4 x − 2 ) We will use the distributive property (also known as the FOIL method) to expand this product.
Expanding the Product Expanding the product:
First, multiply the first terms: 6 x × 4 x = 24 x 2 .
Next, multiply the outer terms: 6 x × − 2 = − 12 x .
Then, multiply the inner terms: 7 × 4 x = 28 x .
Finally, multiply the last terms: 7 × − 2 = − 14 .
So, we have: h ( x ) = 24 x 2 − 12 x + 28 x − 14
Simplifying the Expression Now, we combine like terms: h ( x ) = 24 x 2 + ( − 12 x + 28 x ) − 14 h ( x ) = 24 x 2 + 16 x − 14
Final Answer Therefore, h ( x ) = 24 x 2 + 16 x − 14 . Comparing this to the given options, we see that it matches the third option.
Examples
Understanding how to multiply functions is useful in many real-world scenarios. For example, if you're running a business, you might model your revenue f ( x ) and your costs g ( x ) as functions of the number of items you sell, x . Then, the profit margin could be expressed as a new function h ( x ) = f ( x ) − g ( x ) . Similarly, if you want to calculate the area of a rectangle where the length and width are defined by functions, you would multiply those functions together to get a new function representing the area.
