What is the inverse function of f(x)=3^x+2?

Mar 27, 2017

$y = \frac{\ln \left(x - 2\right)}{\ln} \left(3\right)$

Explanation:

Set $\text{ } y = {3}^{x} + 2$

The $y$ is called the 'dependant variable' and the $x$ is called the 'independent variable'. That is because we may assign any value we choose to $x$ and the value of $y$ depends on what we assign to $x$

We need to make $x$ the dependant variable.

I am assuming you know the shortcut methods of manipulation.

Move the 2 to the other side

$y - 2 = {3}^{x}$

Take logs of both sides ( I choose ln)

$\ln \left(y - 2\right) = \ln \left({3}^{x}\right)$

This is the same as:

$\ln \left(y - 2\right) = x \ln \left(3\right)$

Move $\ln \left(3\right)$ to the other side

$\frac{\ln \left(y - 2\right)}{\ln} \left(3\right) = x$

Where there is a $y$ write $x$ and where there is a $x$ write $y$

$\frac{\ln \left(x - 2\right)}{\ln} \left(3\right) = y$
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Notice that

$y = \frac{\ln \left(x - 2\right)}{\ln} \left(3\right)$

is a reflection about $y = x$

of $y = {3}^{x} + 2$