Answer Key 11.3
[latexpage]
- \(\begin{array}{rrl}
\\ \\ \\
(g\circ f)(x)&=&-(\sqrt[5]{-x-3})^5-3 \\
&=&-(-x-3)-3 \\
&=&x+3-3 \\
&=&x\hspace{0.75in}\text{Inverse}
\end{array}\) - \(\begin{array}{rrl}
\\ \\ \\
(g\circ f)(x)&=&4-\left(\dfrac{4}{x}\right) \\ \\
&=&4-\dfrac{4}{x}\hspace{0.5in}\text{Not inverse}
\end{array}\) - \(\begin{array}{rrl}
\\ \\ \\ \\
(g\circ f)(x)&=&-10\left(\dfrac{x-5}{10}\right)+5 \\ \\
&=& -x+5+5\\ \\
&=&-x+10\hspace{0.75in}\text{Not inverse}
\end{array}\) - \(\begin{array}{rrl}
\\ \\ \\ \\ \\ \\ \\
(f\circ g)(x)&=&\dfrac{(10x+5)-5}{10} \\ \\
&=&\dfrac{10x+5-5}{10} \\ \\
&=&\dfrac{10x}{10} \\ \\
&=&x\hspace{0.75in}\text{Inverse}
\end{array}\) - \(\begin{array}{rrl}
\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
(f\circ g)(x)&=&\dfrac{-2}{\dfrac{3x+2}{x+2}+3} \\ \\
&=& \dfrac{-2(x+2)}{3x+2+3(x+2)}\\ \\
&=& \dfrac{-2x-4}{3x+2+3x+6}\\ \\
&=& \dfrac{-2x-4}{6x+8}\\ \\
&=& \dfrac{-x-2}{3x+4}\hspace{0.75in}\text{Not inverse}
\end{array}\) - \(\begin{array}{rrl}
\\ \\ \\ \\ \\ \\ \\ \\ \\ \\
(f\circ g)&=&\dfrac{-\left(\dfrac{-2x+1}{-x-1}\right)-1}{\dfrac{-2x+1}{-x-1}-2} \\ \\
&=&\dfrac{-(-2x+1)-1(-x-1)}{-2x+1-2(-x-1)} \\ \\
&=&\dfrac{2x-1+x+1}{-2x+1+2x+2} \\ \\
&=&\dfrac{3x}{3} \\ \\
&=&x\hspace{0.75in}\text{Inverse}
\end{array}\) - \(\begin{array}{rrcrr}
\\ \\ \\ \\ \\ \\
y&=&(x-2)^5&+&3 \\
x&=&(y-2)^5&+&3 \\
-3&&&-&3 \\
\midrule
x-3&=&(y-2)^5&& \\
\sqrt[5]{x-3}&=&y-2&& \\ \\
y&=&\sqrt[5]{x-3}&+&2
\end{array}\) - \(\begin{array}{rrcrr}
\\ \\ \\ \\ \\ \\
y&=&\sqrt[3]{x+1}&+&2 \\
x&=&\sqrt[3]{y+1}&+&2 \\
-2&&&-&2 \\
\midrule
x-2&=&\sqrt[3]{y+1}&& \\
(x-2)^3&=&y+1&& \\ \\
y&=&(x-2)^3&-&1
\end{array}\) - \(\begin{array}{rrl}
\\ \\ \\ \\ \\ \\ \\ \\
y&=&\dfrac{4}{x+2} \\ \\
x&=&\dfrac{4}{y+2} \\ \\
y+2&=&\dfrac{4}{x} \\ \\
y&=&\dfrac{4}{x}-2
\end{array}\) - \(\begin{array}{rrl}
\\ \\ \\ \\ \\ \\ \\ \\
y&=&\dfrac{3}{x-3} \\ \\
x&=&\dfrac{3}{y-3} \\ \\
y-3&=&\dfrac{3}{x} \\ \\
y&=&\dfrac{3}{x}+3
\end{array}\) - \(\begin{array}{rrl}
\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
y&=&\dfrac{-2x-2}{x+2} \\ \\
x&=&\dfrac{-2y-2}{y+2} \\ \\
x(y+2)&=&-2y-2 \\
xy+2x&=&-2y-2 \\
-xy+2&&-xy+2 \\
\midrule
2x+2&=&-2y-xy \\
2x+2&=&y(-2-x) \\ \\
y&=&\dfrac{2x+2}{-2-x} \\ \\
y&=&-\dfrac{2x+2}{2+x}
\end{array}\) - \(\begin{array}{rrl}
\\ \\ \\ \\ \\
y&=&\dfrac{9+x}{3} \\ \\
x&=&\dfrac{9+y}{3} \\ \\
3x&=&9+y \\
y&=&3x-9
\end{array}\) - \(\begin{array}{rrl}
\\ \\ \\ \\ \\
y&=&\dfrac{10-x}{5} \\ \\
x&=&\dfrac{10-y}{5} \\ \\
5x&=&10-y \\
y&=&10-5x
\end{array}\) - \(\begin{array}{rrl}
\\ \\ \\ \\ \\ \\ \\ \\ \\
y&=&\dfrac{5x-15}{2} \\ \\
x&=&\dfrac{5y-15}{2} \\ \\
5y-15&=&2x \\ \\
5y&=&2x+15 \\ \\
y&=&\dfrac{2x+15}{5}
\end{array}\) - \(\begin{array}{rrl}
\\ \\ \\ \\ \\ \\
y&=&-(x-1)^3 \\
x&=&-(y-1)^3 \\
\sqrt[3]{x}&=&-(y-1) \\
\sqrt[3]{x}&=&-y+1 \\
-y&=&\sqrt[3]{x}-1 \\ \\
y&=&1-\sqrt[3]{x}
\end{array}\) - \(\begin{array}{rrl}
\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
y&=&\dfrac{12-3x}{4} \\ \\
x&=&\dfrac{12-3y}{4} \\ \\
4x&=&12-3y \\ \\
3y&=&12-4x \\ \\
y&=&\dfrac{12-4x}{3} \\ \\
y&=&4-\dfrac{4}{3}x
\end{array}\) - \(\begin{array}{rrl}
\\ \\ \\ \\
y&=&(x-3)^3 \\
x&=&(y-3)^3 \\
\sqrt[3]{x}&=&y-3 \\ \\
y&=&\sqrt[3]{x}+3
\end{array}\) - \(\begin{array}{rrl}
\\ \\ \\ \\ \\
y&=&\sqrt[5]{-x}+2 \\
x&=&\sqrt[5]{-y}+2 \\
x-2&=&\sqrt[5]{-y} \\
(x-2)^5&=&-y \\ \\
y&=&-(x-2)^5
\end{array}\) - \(\begin{array}{rrl}
\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
y&=&\dfrac{x}{x-1} \\ \\
x&=&\dfrac{y}{y-1} \\ \\
x(y-1)&=&y \\ \\
xy-x&=&y \\ \\
y-xy&=&-x \\ \\
y(1-x)&=&-x \\ \\
y&=&\dfrac{-x}{1-x} \\ \\
y&=&\dfrac{x}{x-1}
\end{array}\) - \(\begin{array}{rrl}
\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
y&=&\dfrac{-3-2x}{x+3} \\ \\
x&=&\dfrac{-3-2y}{y+3} \\ \\
x(y+3)&=&-3-2y \\
xy+3x&=&-3-2y \\
+2y-3x&&-3x+2y \\
\midrule
xy+2y&=&-3-3x \\
y(x+2)&=&-3-3x \\ \\
y&=&\dfrac{-3-3x}{x+2} \\ \\
y&=&-\dfrac{3x+3}{x+2}
\end{array}\) - \(\begin{array}{rrl}
\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
y&=&\dfrac{x-1}{x+1} \\ \\
x&=&\dfrac{y-1}{y+1} \\ \\
x(y+1)&=&y-1 \\
xy+x&=&y-1 \\
xy-y&=&-x-1 \\
y(x-1)&=&-x-1 \\ \\
y&=&\dfrac{-x-1}{x-1} \\ \\
y&=&-\dfrac{x+1}{x-1}
\end{array}\) - \(\begin{array}{rrl}
\\ \\ \\ \\ \\ \\ \\ \\ \\ \\
y&=&\dfrac{x}{x+2} \\ \\
x&=&\dfrac{y}{y+2} \\ \\
x(y+2)&=&y \\
xy+2x&=&y \\
2x&=&y-xy \\
2x&=&y(1-x) \\ \\
y&=&\dfrac{2x}{1-x}
\end{array}\)
