Hey I am having some trouble finding the inverse function of
f(x) = (2^x)-1
and also I need help with proving the functions are mutually inverse can you please show me how to do it.
\[ \text{If }y = \left(2^x\right)^{-1}\\ \text{then for the inverse, after swapping }x\text{ and }y, \]
\begin{align*} x &= \left(2^y\right)^{-1}\\ \frac{1}{x} &= 2^y\\ \log_2 \frac1x &= y \end{align*}
\[ \therefore f^{-1}(x) = \log_2 \frac{1}{y} \]
Ensure that you
know that the exponential and logarithm are inverse functions. Because the logarithm is
defined to be the inverse function to the exponential, the statements \(a^x = b\) and \(x = \log_a b\) are equivalent. i.e. we can go between one another.
Also, because they are inverses, the mutual inverses property is
assumed to hold here. That is, we can
always assume that \( a^{\log_a x} = x\) for \(x > 0\), and \(\log_a (a^x) = x\) for all real \(x\).