ANSWER
[tex]{f}^{ - 1} (x) = ln(x + 4) - 3[/tex]
EXPLANATION
We want to find the equation of the inverse of
[tex]y = {e}^{x + 3} - 4[/tex]
First, we need to interchange x and y.
[tex]x= {e}^{y + 3} - 4[/tex]
Solve for y,
[tex]x + 4={e}^{y + 3} [/tex]
Take the natural logarithm of both sides,
[tex] ln(x + 4)= ln{e}^{y + 3} [/tex]
Simplify
[tex] ln(x + 4)= y + 3[/tex]
[tex]ln(x + 4) - 3= y [/tex]
Hence the inverse function is,
[tex] {f}^{ - 1} (x) = ln(x + 4) - 3[/tex]
