Perform the indicated operation.
1)
[tex]
\begin{array}{l}
h(x)-9 x \\
g(x)=4 x+2 \\
\text { Find }(h \cdot g)(4)
\end{array}
[/tex]
2)
[tex]
\begin{array}{l}
h(x)-2 z \\
g(x)-8 x^3+6 x^2 \\
\text { Find }\left(\frac{f}{h}\right)(x)
\end{array}
[/tex]
3)
[tex]
\begin{array}{l}
h(x)=10 x \\
g(x)=30 x^3+40 x^3 \\
\text { Find }\left(\frac{q}{h}\right)(1)
\end{array}
[/tex]
4)
[tex]
\begin{array}{l}
h(x)=-4 x \\
g(x)=2 x-3 \\
\text { Find }(h \cdot g)(x)
\end{array}
[/tex]
5)
[tex]
\begin{array}{l}
h(x)=7 z \\
g(z)-8 x+3 \\
\text { Find }(h \cdot g)(1)
\end{array}
[/tex]
6)
[tex]
\begin{array}{l}
h(x)=3 x \\
g(x)-6 x+4 \\
\text { Find }(h \cdot g)(3)
\end{array}
[/tex]
7)
[tex]
\begin{array}{l}
h(x)=-2 z \\
g(z)=4 x-3 \\
\text { Find }(h, g)(z)
\end{array}
[/tex]
8)
[tex]
\begin{array}{l}
h(x)=5 z \\
g(z)--10 x^3+10 z^2 \\
\text { Find }\left(\frac{z}{h}\right)(z)
\end{array}
[/tex]
9)
[tex]
\begin{array}{l}
h(x)=10 x \\
g(x)=8 x+3 \\
\text { Find }(h . g)(x)
\end{array}
[/tex]
10)
[tex]
\begin{array}{l}
h(x)=9 z \\
g(x)=-27 x^3+36 z^3 \\
\text { Find }\left(\frac{g}{h}\right)(z)
\end{array}
[/tex]
11) [tex]h(x)=2 z[/tex]
[tex]
\begin{array}{l}
g(x)=-4 x^3+8 x^2 \\
\text { Find }\left(\frac{9}{h}\right)(3)
\end{array}
[/tex]
12)
[tex]
\begin{array}{l}
h(x)=-5 z \\
g(z)=6 z-2 \\
\text { Find }(h \cdot g)(z)
\end{array}
[/tex]
13)
[tex]
\begin{array}{l}
h(x)=4 x \\
g(x)=16 x^3+16 x^3 \\
\text { Find }\left(\frac{g}{h}\right)(3)
\end{array}
[/tex]
14)
[tex]
\begin{array}{l}
h(x)=6 z \\
g(x)=-18 x^3+12 z^3 \\
\text { Find }\left(\frac{7}{h}\right)(4)
\end{array}
[/tex]
15) [tex]h(x)=10 x[/tex]
[tex](2) =20 x^3+20 x^2[/tex]
[tex]=z^{\prime}=\text { ind }\left(\frac{g}{b}\right)(z)[/tex]
16)
[tex]
\begin{array}{l}
h(x)=8 z \\
g(x)=-24 x^3+32 z^2 \\
\text { Find }\left(\frac{z}{h}\right)(2)
\end{array}
[/tex]
Answer (2)
Evaluate h ( 4 ) : h ( 4 ) = 9 \tcdot 4 = 36 .
Evaluate g ( 4 ) : g ( 4 ) = 4 \tcdot 4 + 2 = 18 .
Multiply h ( 4 ) and g ( 4 ) : ( h \tcdot g ) ( 4 ) = 36 \tcdot 18 .
Calculate the final result: ( h \tcdot g ) ( 4 ) = 648 .
Explanation
Problem Analysis We are given two functions, h ( x ) = 9 x and g ( x ) = 4 x + 2 . We are asked to find ( h \tcdot g ) ( 4 ) , which means we need to find the product of the two functions evaluated at x = 4 .
Calculate h(4) First, let's evaluate h ( 4 ) :
h ( 4 ) = 9 ( 4 ) = 36
Calculate g(4) Next, let's evaluate g ( 4 ) :
g ( 4 ) = 4 ( 4 ) + 2 = 16 + 2 = 18
Multiply h(4) and g(4) Now, we multiply the two results: ( h \tcdot g ) ( 4 ) = h ( 4 ) \tcdot g ( 4 ) = 36 \tcdot 18
Final Calculation Let's calculate the final result: 36 \tcdot 18 = 648
Final Answer Therefore, ( h \tcdot g ) ( 4 ) = 648 .
Examples
Understanding function operations like multiplication is essential in various fields. For instance, in economics, if h ( x ) represents the number of workers and g ( x ) represents the average productivity per worker at a given investment level x , then ( h \tcdot g ) ( x ) gives the total production. Evaluating this at a specific investment level, like x = 4 , helps economists determine the total output at that level. This concept extends to other areas, such as calculating total revenue based on the number of items sold and the price per item.
To find ( h ⋅ g ) ( 4 ) , we evaluate h ( 4 ) and g ( 4 ) and then multiply these results. After calculations, we find that ( h ⋅ g ) ( 4 ) = 648 .
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