Solved on Feb 19, 2024

Find the inverse function f1(x)f^{-1}(x) if f(x)=ex+12f(x) = e^{x+1} - 2.

STEP 1

Assumptions
1. The function given is f(x)=ex+12 f(x) = e^{x+1} - 2 .
2. We are looking for the inverse function f1(x) f^{-1}(x) .
3. The domain of f f is such that the inverse function exists.

STEP 2

To find the inverse function f1(x) f^{-1}(x) , we need to express x x in terms of y y by replacing f(x) f(x) with y y .
y=ex+12 y = e^{x+1} - 2

STEP 3

Next, we isolate the exponential term on one side of the equation.
y+2=ex+1 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 x .
ln(y+2)=ln(ex+1) \ln(y + 2) = \ln(e^{x+1})

STEP 5

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

STEP 6

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

STEP 7

Now that we have x x in terms of y y , we can write the inverse function by replacing y y with x x .
f1(x)=ln(x+2)1 f^{-1}(x) = \ln(x + 2) - 1
The correct answer is f1(x)=ln(x+2)1 f^{-1}(x) = \ln(x + 2) - 1 , which corresponds to one of the given options.
Solution: f1(x)=ln(x+2)1 f^{-1}(x) = \ln(x + 2) - 1

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