CPM Homework Help : PC3 Problem 3-47

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3-47.

Write the equation of $f^{-1}(x)$ if:

$f\left(x\right)=\frac{x}{x+1}$
Step 1:
$\text{Replace }f(x)\text{ with }y:\ y=\frac{x}{x+1}$
Step 2:
$\text{Switch }x\text{ and }y: x = \frac{y}{y + 1}$
Step 3:
Multiply both sides by $(y+1)$ .
$(\textit{y} + 1)\textit{x} = \textit{y}$
Step 4:
$yx + x = y$
Step 5:
Isolate $y$ on the left side.
$yx - y = -x$
Step 6:
Factor out $y$ .
$y(x - 1) = -x$
Step 7:
Divide both sides by $\left(x-1\right)$ .
$y = \frac{-\textit{x}}{\textit{x} - 1}\rightarrow\:\textit{y} = \frac{\textit{x}}{1-\textit{x}}$

$f\left(x\right)=5\sqrt[3]{x-2}$

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