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Let $f(x)=x+2$ and $g(x)=x/3.$ Also denote the inverses to these functions as $f^{-1}$ and $g^{-1}.$ Compute \[f(g^{-1}(f^{-1}(f^{-1}(g(f(19)))))).\]

 Nov 16, 2014

Best Answer 

 #1
avatar+21860 
+5

$$\small{\text{
$
\begin{array}{rcl}
f(x)=x+2 \\
f^{-1}(x) &=& ? \\
y & =& x+2 \quad | \quad x\leftrightarrow y \\
x & = & y+2 \\
y & = & x-2 \\
f^{-1}(x) & = & x-2 \\
\hline
\\
g(x)=\dfrac{x}{3} \\ \\
g^{-1}(x) &=& ? \\
y & =& \dfrac{x}{3} \quad | \quad x\leftrightarrow y \\ \\
x & = & \dfrac{y}{3} \\ \\
y & = & 3x \\
g^{-1}(x) & = & 3x \\
\hline
\end{array}
$
}}$$

$$\small{\text{
$
\begin{array}{rcccl}
f(19) & = & 19 + 2 & = & 21 \\
g(21) & = & \frac{21}{3} & = & 7 \\
f^{-1}(7) & = & 7-2 & = & 5 \\
f^{-1}(5) & = & 5-2 & = & 3 \\
g^{-1}(3) & = & 3*3 & = & 9 \\
f(9) & = & 9 + 2 & = & 11 \\
\end{array}
$
}}$$

$$f(g^{-1}(f^{-1}(f^{-1}(g(f(19)))))) = 11$$

.
 Nov 17, 2014
 #1
avatar+21860 
+5
Best Answer

$$\small{\text{
$
\begin{array}{rcl}
f(x)=x+2 \\
f^{-1}(x) &=& ? \\
y & =& x+2 \quad | \quad x\leftrightarrow y \\
x & = & y+2 \\
y & = & x-2 \\
f^{-1}(x) & = & x-2 \\
\hline
\\
g(x)=\dfrac{x}{3} \\ \\
g^{-1}(x) &=& ? \\
y & =& \dfrac{x}{3} \quad | \quad x\leftrightarrow y \\ \\
x & = & \dfrac{y}{3} \\ \\
y & = & 3x \\
g^{-1}(x) & = & 3x \\
\hline
\end{array}
$
}}$$

$$\small{\text{
$
\begin{array}{rcccl}
f(19) & = & 19 + 2 & = & 21 \\
g(21) & = & \frac{21}{3} & = & 7 \\
f^{-1}(7) & = & 7-2 & = & 5 \\
f^{-1}(5) & = & 5-2 & = & 3 \\
g^{-1}(3) & = & 3*3 & = & 9 \\
f(9) & = & 9 + 2 & = & 11 \\
\end{array}
$
}}$$

$$f(g^{-1}(f^{-1}(f^{-1}(g(f(19)))))) = 11$$

heureka Nov 17, 2014

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