Question

How do you find the inverse of `y=e^(3x+1)`?

Answer 1

If `z_x` is the inverse of `y_x =e^(3x+1)`
then `z_x = (ln(x)-1)/3`
(with some restrictions)

Explanation:

If `z_x` is the inverse of `y_x`
`color(white)("XXX")y_(z_x) = x` (by definition of inverse)

but
`color(white)("XXX")y_(z_x) = e^(3z_x+1)` (by definition of `y_x`)

therefore
`color(white)("XXX")e^(3z_x+1) = x`

and taking the natural log of both sides
`color(white)("XXX")3z_x+1=ln(x)`

`rArr`
`color(white)("XXX")z_x= (ln(x)-1)/3`

Note however that for certain values of `x`,
`y_x` generates non-reversible values (because of limitations on argument values for the `ln` function)