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Solved on Feb 19, 2024

Find the inverse function $f^{-1}(x)$ if $f(x) = e^{x+1} - 2$ .

STEP 1

Assumptions
1. The function given is $f(x) = e^{x+1} - 2$ .
2. We are looking for the inverse function $f^{-1}(x)$ .
3. The domain of $f$ is such that the inverse function exists.

STEP 2

To find the inverse function $f^{-1}(x)$ , we need to express $x$ in terms of $y$ by replacing $f(x)$ with $y$ .
$y = e^{x+1} - 2$

STEP 3

Next, we isolate the exponential term on one side of the equation.
$y + 2 = e^{x+1}$

STEP 4

Now, we take the natural logarithm (ln) of both sides to get rid of the exponential and solve for $x$ .
$\ln(y + 2) = \ln(e^{x+1})$

STEP 5

Use the property of logarithms that states $\ln(e^a) = a$ to simplify the right side of the equation.
$\ln(y + 2) = x + 1$

STEP 6

Subtract 1 from both sides to solve for $x$ .
$\ln(y + 2) - 1 = x$

STEP 7

Now that we have $x$ in terms of $y$ , we can write the inverse function by replacing $y$ with $x$ .
$f^{-1}(x) = \ln(x + 2) - 1$
The correct answer is $f^{-1}(x) = \ln(x + 2) - 1$ , which corresponds to one of the given options.
Solution: $f^{-1}(x) = \ln(x + 2) - 1$

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