Question

Inverse Function Question

Determine 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{(x)/(5)}`

Now take their composition,

`(g\circ f)(x)=g(f(x))=(\ln(x))/(5)`

Let
`y=(\ln(x))/(5)`, now again find the inverse,

`x=(\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 :)