Question

How do I find the inverse function of `f(x)=e^(3x)-4`? And what is it's range ?

Answer 1

To find the inverse algebraically, switch the x and ys.

`x = e^(3y) -4`

`x +4 = e^(3y)`

`ln(x+ 4) = ln(e^(3y))`

`ln(x + 4) = 3y`

`y = f^-1(x) = 1/3ln(x +4)`

Since the inverse of the function is the original function reflected over the line `y =x`, the domain of the original function becomes the range of the inverse and vice versa. Since `y = e^(3x) -4` has a domain of all the real numbers, `y = 1/3ln(x+ 4)` will have a range of all the real numbers.

Hopefully this helps!