f(x)=e^x+3+5, Need Domain(Which I couldn't find one) Where is the inverse?

Question

anonymous · Answer

I did this problem, couldn't find a domain, but teacher states I should have one.

anonymous · Answer

$$f(x)=e ^{(x+3)}+5$$Exponential functions have an unrestricted domain, so$$dom(f)=\mathbb{R}=(-\infty,\infty)$$

anonymous · Answer

The function is one-to-one, so it has an inverse.  To find the inverse, use the four step method.  

1)  Let f(x)=y, so$$f(x)=e ^{(x+3)}+5=y$$2)  f^-1(x) is implied by$$x=e ^{(y+3)}+5$$(here, we switch the x and y)

3) Solve for y$$x=e ^{y+3}+5\rightarrow  x-5=e ^{y+3} \rightarrow \ln (x-5)=y+3\rightarrow \ln (x-5)-3=y$$4) Let y=f^-1(x), so $$\ln (x-5)-3=y=f ^{-1}(x)$$