Answer Key 11.2
[latexpage]
- \(\begin{array}{ll}
\\ \\
\begin{array}{rrl}
g(3)&=&(3)^3+5(3)^2 \\
&=&27+45 \\
&=&72
\end{array}
&\hspace{0.25in}
\begin{array}{rrl}
f(3)&=&2(3)+4 \\
&=&6+4 \\
&=&10 \\
\end{array}
\end{array}\)\(g(3)+f(3)=72+10=82\)
- \(\begin{array}{ll}
\\ \\
\begin{array}{rrl}
\\
f(-4)&=&-3(-4)^2+3(-4) \\
&=&-3(16)-12 \\
&=&-48-12 \\
&=&-60
\end{array}
&\hspace{0.25in}
\begin{array}{rrl}
g(-4)&=&2(-4)+5 \\
&=&-8+5 \\
&=&-3
\end{array}
\end{array}\)\(\dfrac{f(-4)}{g(-4)}=\dfrac{-60}{-3}=20\)
- \(\begin{array}{ll}
\\ \\
\begin{array}{rrl}
g(5)&=&-4(5)+1 \\
&=&-20+1 \\
&=&-19
\end{array}
& \hspace{0.25in}
\begin{array}{rrl}
h(5)&=&-2(5)-1 \\
&=&-10-1 \\
&=& -11
\end{array}
\end{array}\)\(g(5)+h(5)=-19-11=-30\)
- \(\begin{array}{ll}
\\ \\
\begin{array}{rrl}
g(2)&=&3(2)+1 \\
&=&6+1 \\
&=&7
\end{array}
& \hspace{0.25in}
\begin{array}{rrl}
\\
f(2)&=&(2)^3+3(2)^2 \\
&=&8+3\cdot 4 \\
&=&8+12 \\
&=&20
\end{array}
\end{array}\)\(g(2)\cdot f(2)=7\cdot 20=140\)
- \(\begin{array}{ll}
\\
\begin{array}{rrl}
g(1)&=&1-3 \\
&=&-2
\end{array}
&\hspace{0.25in}
\begin{array}{rrl}
\\
h(1)&=&-3(1)^3+6(1) \\
&=&-3+6 \\
&=&3
\end{array}
\end{array}\)\(g(1)+h(1)=-2+3=1\)
- \(\begin{array}{ll}
\\ \\
\begin{array}{rrl}
g(-6)&=&(-6)^2-2 \\
&=&36-2 \\
&=&34
\end{array}
& \hspace{0.25in}
\begin{array}{rrl}
h(-6)&=&2(-6)+5 \\
&=&-12+5 \\
&=&-7
\end{array}
\end{array}\)\(g(-6)+h(-6)=34-7=27\)
- \(\begin{array}{ll}
\\
\begin{array}{rrl}
h(0)&=&\cancel{2(0)}-1 \\
&=&-1
\end{array}
& \hspace{0.25in}
\begin{array}{rrl}
g(0)&=&\cancel{3(0)}-5 \\
&=&-5
\end{array}
\end{array}\)\(\dfrac{h(0)}{g(0)}=\dfrac{-1}{-5}=\dfrac{1}{5}\)
- \((g+h)=
\begin{array}{rrrr}
\\ \\
&3a&-&2 \\
+&4a&-&2 \\
\midrule
&7a&-&4
\end{array}\hspace{0.25in}
\begin{array}{rrl}
\\ \\
(g+h)(10)&=&7(10)-4 \\
&=&70-4 \\
&=&66
\end{array}\) - \((g+f)=
\begin{array}{rrrr}
\\ \\
&3a&+&3 \\
+&2a&-&2 \\
\midrule
&5a&+&1
\end{array}\hspace{0.25in}
\begin{array}{rrl}
\\ \\
(g+f)(9)&=&5(9)+1 \\
&=&45+1 \\
&=&46
\end{array}\) - \((g-h)=
\begin{array}{r}
\\ \\
4x+3 \\
- \hspace{0.42in} (x^3-2x^2) \\
\midrule
-x^3+2x^2+4x+3
\end{array}\hspace{0.25in}
\begin{array}{rrl}
\\ \\
(g-h)(-1)&=&-(-1)^3+2(-1)^2+4(-1)+3 \\
&=&1+2-4+3 \\
&=&2
\end{array}\) - \((g-f)=
\begin{array}{rrrr}
\\ \\
&x&+&3 \\
-&(-x&+&4) \\
\midrule
&2x&-&1
\end{array}\hspace{0.25in}
\begin{array}{rrl}
\\ \\
(g-f)(3)&=&2(3)-1 \\
&=&6-1 \\
&=&5
\end{array}\) - \((g-f)=
\begin{array}{rrrrrr}
\\ \\
&x^2&&&+&2 \\
-&&&(2x&+&5) \\
\midrule
&x^2&-&2x&-&3
\end{array}\hspace{0.25in}
\begin{array}{rrl}
\\ \\
(g-f)(0)&=&\cancel{(0)^2}-\cancel{2(0)}-3 \\
&=&-3
\end{array}\) - \((f+g)=
\begin{array}{rrrr}
\\ \\
&n&-&5 \\
+&4n&+&2 \\
\midrule
&5n&-&3
\end{array}\hspace{0.25in}
\begin{array}{rrl}
\\ \\
(f+g)(-8)&=&5(-8)-3 \\
&=&-40-3 \\
&=&-43
\end{array}\) - \((h\cdot g)=
\begin{array}{rrrrrr}
\\ \\ \\ \\ \\
&&&t&+&5 \\
\times &&&3t&-&5 \\
\midrule
&3t^2&+&15t&& \\
&&-&5t&-&25 \\
\midrule
&3t^2&+&10t&-&25
\end{array}\hspace{0.25in}
\begin{array}{rrl}
\\ \\
(h\cdot g)(5)&=&3(5)^2+10(5)-25 \\
&=&75+50-25 \\
&=&100
\end{array}\) - \((g\cdot h)=
\begin{array}{rrrr}
\\ \\
&t&-&4 \\
\times &&&2t \\
\midrule
&2t^2&-&8t
\end{array}\hspace{0.25in}
\begin{array}{rrl}
\\ \\
(g\cdot h)(3t)&=&2(3t)^2-8(3t) \\
&=&2(9t^2)-24t \\
&=&18t^2-24t
\end{array}\) - \(\dfrac{g(n)}{f(n)}=\dfrac{n^2+5}{2n+5}\hspace{0.25in} \text{Does not reduce}\)
- \(\dfrac{g}{f}=\dfrac{-2a+5}{3a+5}\hspace{0.3in}\left(\dfrac{g}{f}\right)(a^2)=\dfrac{-2a^2+5}{3a^2+5}\hspace{0.25in} \text{Does not reduce}\)
- \(h(n)+g(n)=
\begin{array}{rrrrrr}
\\ \\
&n^3&+&4n&& \\
+&&&4n&+&5 \\
\midrule
&n^3&+&8n&+&5
\end{array}\) - \(\begin{array}{rrl}
\\
g(n^2)&=&(n^2)^2-4(n^2) \\
&=&n^4-4n^2
\end{array}\hspace{0.25in}
h(n^2)=n^2-5\)\(g(n^2)\cdot h(n^2)=
\begin{array}{rrrrrr}
\\ \\ \\
&&&n^4&-&4n^2 \\
\times&&&n^2&-&5 \\
\midrule
&n^6&-&4n^4&& \\
&&-&5n^4&+&20n^2 \\
\midrule
&n^6&-&9n^4&+&20n^2 \\
\end{array}\) - \((g\cdot h)=
\begin{array}{rrrrrr}
\\ \\ \\
&&&n&+&5 \\
\times &&&2n&-&5 \\
\midrule
&2n^2&+&10n&& \\
&&-&5n&-&25 \\
\midrule
&2n^2&+&5n&-&25
\end{array}\hspace{0.25in}
\begin{array}{rrl}
(g\cdot h)(-3n)&=&2(-3n)^2+5(-3n)-25 \\
&=&2(9n^2)-15n-25 \\
&=&18n^2-15n-25
\end{array}\) - \(\begin{array}{ll}
\\ \\
\begin{array}{rrl}
(f\circ g)&=&-4(4x+3)+1 \\
&=&-16x-12+1 \\
&=&-16x-11
\end{array}
&\hspace{0.25in}
\begin{array}{rrl}
(f\circ g)(9)&=&-16(9)-11 \\
&=&-144-11 \\
&=&-155
\end{array}
\end{array}\) - \(\begin{array}{ll}
\\ \\
\begin{array}{rrl}
(h\circ g)&=&3(a+1)+3 \\
&=&3a+3+3 \\
&=&3a+6
\end{array}
&\hspace{0.25in}
\begin{array}{rrl}
(h\circ g)(5)&=&3(5)+6 \\
&=&15+6 \\
&=&21
\end{array}
\end{array}\) - \(\begin{array}{ll}
\\ \\
\begin{array}{rrl}
(g\circ h)&=&(x^2-1)+4 \\
&=&x^2-1+4 \\
&=&x^2+3
\end{array}
&\hspace{0.25in}
\begin{array}{rrl}
(g\circ h)(10)&=&(10)^2+3 \\
&=&100+3 \\
&=&103
\end{array}
\end{array}\) - \(\begin{array}{ll}
\\ \\
\begin{array}{rrl}
(f\circ g)&=&-4(n+4)+2 \\
&=&-4n-16+2 \\
&=&-4n-14
\end{array}
&\hspace{0.25in}
\begin{array}{rrl}
(f\circ g)(9)&=&-4(9)-14 \\
&=&-36-14 \\
&=&-50
\end{array}
\end{array}\) - \(\begin{array}{ll}
\begin{array}{rrl}
\\
(g\circ h)&=&2(2x^3+4x^2)-4 \\
&=&4x^3+8x^2-4
\end{array}
&\hspace{0.25in}
\begin{array}{rrl}
\\ \\
(g\circ h)(3)&=&4(3)^3+8(3)^2-4 \\
&=&108+72-4 \\
&=&176
\end{array}
\end{array}\) - \(\begin{array}{rrl}
\\ \\
(g\circ h)&=&(4x+4)^2-5(4x+4) \\
&=&16x^2+32x+16-20x-20 \\
&=&16x^2+12x-4
\end{array}\) - \(\begin{array}{rrl}
\\
(f\circ g)&=&-2(4a)+2 \\
&=&-8a+2
\end{array}\) - \(\begin{array}{rrl}
\\ \\
(g\circ f)&=&4(x^3-1)+4 \\
&=&4x^3-4+4 \\
& =&4x^3
\end{array}\) - \(\begin{array}{rrl}
\\ \\
(g\circ f)&=&-(2x-3)+5 \\
&=&-2x+6+5 \\
&=&-2x+11
\end{array}\) - \(\begin{array}{rrl}
\\ \\
(f\circ g)&=&4(-4t-2)+3 \\
&=&-16t-8+3 \\
&=&-16t-5
\end{array}\)
