Find the inverse of the logarithmic function $f$ .
$f (x) = ln (x + 2) - 3$

e^{x + 3} - 2

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e^{x + 1} - 2

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e^{x + 2} - 3

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e^{x - 2} + 3

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Solution

The correct option is A $e^{x + 3} - 2$
Let $y = f (x) = ln (x + 2) - 3$ and now solve for $x$ to find the inverse of the given function.
$y = ln (x + 2) - 3$
$ln (x + 2) = y + 3$
$x + 2 = e^{y + 3}$
$x = e^{y + 3} - 2$
Therefore, the inverse of the function in terms of $x$ is $e^{x + 3} - 2$ .

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f

f (x) = e^{x - 1} + 3

f (x) = \frac{e^{x} - e^{- x}}{e^{x} + e^{- x}} + 2

\frac{(e^{x} + 2)}{(e^{x} - 1) (2 e^{x} - 3)} = \frac{- 3}{e^{x} - 1} + \frac{B}{2 e^{x} - 3}

The inverse of the function $f (x) = \frac{e^{x} - e^{- x}}{e^{x} + e^{- x}} + 2$ is given by

1. f (x) = x^{3}, x ϵ R

2. f (x) = sin x, - π / 2 < x < π / 2

3. f (x) = e^{x}, x ϵ R

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