QuestionFind the inverse of the function $g(x) = \frac{2}{x+2} - 1$ .

Studdy Solution

STEP 1

1. We are given the function $g(x) = \frac{2}{x+2} - 1$ .
2. We need to find the inverse of this function, denoted as $g^{-1}(x)$ .

STEP 2

1. Replace $g(x)$ with $y$ to facilitate solving for the inverse.
2. Solve the equation for $x$ in terms of $y$ .
3. Swap $x$ and $y$ to find the inverse function.
4. Simplify the expression to get the inverse function $g^{-1}(x)$ .

STEP 3

Replace $g(x)$ with $y$ :
$y = \frac{2}{x+2} - 1$

STEP 4

Solve for $x$ in terms of $y$ .
First, isolate the fraction by adding 1 to both sides:
$y + 1 = \frac{2}{x+2}$

STEP 5

Cross-multiply to eliminate the fraction:
$(y + 1)(x + 2) = 2$

STEP 6

Expand the left side:
$yx + 2y + x + 2 = 2$

STEP 7

Rearrange the equation to solve for $x$ :
$yx + x = 2 - 2y$
Factor out $x$ from the left side:
$x(y + 1) = 2 - 2y$

STEP 8

Solve for $x$ :
$x = \frac{2 - 2y}{y + 1}$

STEP 9

Swap $x$ and $y$ to find the inverse function:
$y = \frac{2 - 2x}{x + 1}$

STEP 10

The inverse function is:
$g^{-1}(x) = \frac{2 - 2x}{x + 1}$
The inverse of the function $g(x)$ is:
$\boxed{g^{-1}(x) = \frac{2 - 2x}{x + 1}}$

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