Question

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

Answer 1

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

Answer 2

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

Answer 3

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 \[\ln (x-5)-3=y=f ^{-1}(x)\]