Question

Inverse Function QuestionDetermine the expression of f^-1(x) for f(x)=e^x

Answer 1

First, find the inverse of f,

$y=e^x$

$x=e^y$

Now take the natural logarithm on both sides,

$\ln x=\ln e^y\implies f^{-1}(x)=\boxed{\ln(x)}$

Second, find the inverse of g,

$y=5x\implies g^{-1}(x)=\boxed{\frac{x}{5}}$

Now take their composition,

$(g\circ f)(x)=g(f(x))=\frac{\ln(x)}{5}$

Let $y=\frac{\ln(x)}{5}$, now again find the inverse,

$x=\frac{\ln(y)}{5}$

$5x=\ln y$

exponentiate both sides to base e,

$e^{5x}=e^{\ln y}\implies (g\circ f)^{-1}(x)=\boxed{e^{5x}}$

Hope this helps :)